(4+y^2)dx+(9+x^2)dy=0

{ "query": { "display": "$$\\left(4+y^{2}\\right)dx+\\left(9+x^{2}\\right)dy=0$$", "symbolab_question": "ODE#(4+y^{2})dx+(9+x^{2})dy=0" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "ODE", "subTopic": "FirstSeparable", "default": "y=2\\tan(-\\frac{2}{3}\\arctan(\\frac{x}{3})-c_{1})", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\left(4+y^{2}\\right)dx+\\left(9+x^{2}\\right)dy=0:{\\quad}y=2\\tan\\left(-\\frac{2}{3}\\arctan\\left(\\frac{x}{3}\\right)-c_{1}\\right)$$", "input": "\\left(4+y^{2}\\right)dx+\\left(9+x^{2}\\right)dy=0", "steps": [ { "type": "interim", "title": "Solve separable ODE:$${\\quad}y=2\\tan\\left(-\\frac{2}{3}\\arctan\\left(\\frac{x}{3}\\right)-c_{1}\\right)$$", "input": "\\left(4+y^{2}\\right)dx+\\left(9+x^{2}\\right)dy=0", "steps": [ { "type": "definition", "title": "First order separable Ordinary Differential Equation", "text": "A first order separable ODE has the form of $$N\\left(y\\right){\\cdot}y'=M\\left(x\\right)$$" }, { "type": "step", "primary": "Let $$y$$ be the dependent variable. Divide by $$dx$$:", "result": "4+y^{2}+\\left(9+x^{2}\\right)\\frac{dy}{dx}=0" }, { "type": "step", "primary": "Substitute $$\\frac{dy}{dx}$$ with $$y^{\\prime}\\left(x\\right)$$", "result": "4+y^{2}+\\left(9+x^{2}\\right)y^{^{\\prime}}\\left(x\\right)=0" }, { "type": "interim", "title": "Rewrite in the form of a first order separable ODE", "input": "4+y^{2}+\\left(9+x^{2}\\right)y^{\\prime}\\left(x\\right)=0", "result": "-\\frac{1}{y^{2}+4}y^{\\prime}\\left(x\\right)=\\frac{1}{9+x^{2}}", "steps": [ { "type": "step", "primary": "Standard form of a first order separable ODE:", "secondary": [ "$$N\\left(y\\right){\\cdot}y^{\\prime}\\left(x\\right)=M\\left(x\\right)$$" ] }, { "type": "step", "result": "4+y^{2}+\\left(9+x^{2}\\right)y^{^{\\prime}}\\left(x\\right)=0" }, { "type": "step", "primary": "Subtract $$4+y^{2}$$ from both sides", "result": "4+y^{2}+\\left(9+x^{2}\\right)y^{^{\\prime}}\\left(x\\right)-\\left(4+y^{2}\\right)=0-\\left(4+y^{2}\\right)" }, { "type": "step", "primary": "Simplify", "result": "\\left(9+x^{2}\\right)y^{^{\\prime}}\\left(x\\right)=-y^{2}-4" }, { "type": "step", "primary": "Divide both sides by $$\\left(9+x^{2}\\right)$$", "result": "\\frac{\\left(9+x^{2}\\right)y^{^{\\prime}}\\left(x\\right)}{9+x^{2}}=\\frac{-y^{2}-4}{9+x^{2}}" }, { "type": "step", "primary": "Simplify", "result": "y^{^{\\prime}}\\left(x\\right)=\\frac{-y^{2}-4}{9+x^{2}}" }, { "type": "step", "primary": "Divide both sides by $$-y^{2}-4$$", "result": "\\frac{y^{^{\\prime}}\\left(x\\right)}{-y^{2}-4}=\\frac{\\frac{-y^{2}-4}{9+x^{2}}}{-y^{2}-4}" }, { "type": "step", "primary": "Simplify", "result": "-\\frac{y^{^{\\prime}}\\left(x\\right)}{y^{2}+4}=\\frac{1}{9+x^{2}}" }, { "type": "step", "primary": "Rewrite in standard form", "secondary": [ "$$N\\left(y\\right)=-\\frac{1}{y^{2}+4},\\:{\\quad}M\\left(x\\right)=\\frac{1}{9+x^{2}}$$" ], "result": "-\\frac{1}{y^{2}+4}y^{^{\\prime}}\\left(x\\right)=\\frac{1}{9+x^{2}}" } ], "meta": { "interimType": "Canon First Order Separable ODE 2Eq" } }, { "type": "interim", "title": "Solve $$\\left(-\\frac{1}{y^{2}+4}\\right)y^{\\prime}\\left(x\\right)=\\frac{1}{9+x^{2}}:{\\quad}-\\frac{1}{2}\\arctan\\left(\\frac{y}{2}\\right)=\\frac{1}{3}\\arctan\\left(\\frac{x}{3}\\right)+c_{1}$$", "input": "\\left(-\\frac{1}{y^{2}+4}\\right)y^{\\prime}\\left(x\\right)=\\frac{1}{9+x^{2}}", "steps": [ { "type": "step", "primary": "If$${\\quad}N\\left(y\\right)\\cdot\\:y'=M\\left(x\\right),\\:y'=\\frac{dy}{dx},\\:$$then $$\\int{N\\left(y\\right)}dy=\\int{M\\left(x\\right)}dx$$, up to a constant", "result": "\\int\\:-\\frac{1}{y^{2}+4}dy=\\int\\:\\frac{1}{9+x^{2}}dx" }, { "type": "step", "primary": "Integrate each side of the equation" }, { "type": "interim", "title": "$$\\int\\:\\frac{1}{9+x^{2}}dx=\\frac{1}{3}\\arctan\\left(\\frac{x}{3}\\right)+c_{1}$$", "input": "\\int\\:\\frac{1}{9+x^{2}}dx", "steps": [ { "type": "interim", "title": "Apply Integral Substitution", "input": "\\int\\:\\frac{1}{9+x^{2}}dx", "steps": [ { "type": "definition", "title": "Integral Substitution definition", "text": "$$\\int\\:f\\left(g\\left(x\\right)\\right)\\cdot\\:g'\\left(x\\right)dx=\\int\\:f\\left(u\\right)du,\\:\\quad\\:u=g\\left(x\\right)$$", "secondary": [ "Substitute: $$x=3u$$" ] }, { "type": "step", "primary": "For $$bx^2\\pm\\:a\\:$$substitute $$x=\\frac{\\sqrt{a}}{\\sqrt{b}}u$$<br/>$$a=9,\\:b=1,\\:\\frac{\\sqrt{a}}{\\sqrt{b}}=3\\quad\\Rightarrow\\quad$$substitute $$x=3u$$" }, { "type": "interim", "title": "$$\\frac{dx}{du}=3$$", "input": "\\left(3u\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=3u^{^{\\prime}}" }, { "type": "step", "primary": "Apply the common derivative: $$u^{\\prime}=1$$", "result": "=3\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=3", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Gc+P/Ji46Zyn162LDAa3j8PQlVFV646ejpUuEWqujX2QuIxj9K+Upo9l4tAcwN/gUP9XflYaQHHpNIh9mK9vykvZOgWCK9KtDVNw/QOcQnseEkUz1Tpf3AZfqnNK5WrP" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=3du$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{9+\\left(3u\\right)^{2}}\\cdot\\:3du" }, { "type": "interim", "title": "Simplify $$\\frac{1}{9+\\left(3u\\right)^{2}}\\cdot\\:3:{\\quad}\\frac{1}{3\\left(u^{2}+1\\right)}$$", "input": "\\frac{1}{9+\\left(3u\\right)^{2}}\\cdot\\:3", "steps": [ { "type": "interim", "title": "$$\\frac{1}{9+\\left(3u\\right)^{2}}=\\frac{1}{9+9u^{2}}$$", "input": "\\frac{1}{9+\\left(3u\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$\\left(3u\\right)^{2}=9u^{2}$$", "input": "\\left(3u\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "result": "=3^{2}u^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "$$3^{2}=9$$", "result": "=9u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7dcunpyWu01Fn286nqdr9Oc0ag8T1MwTer44+aCS/ZFA6e/EvvVbAhbFKDdRFDpQ7tI5NPjAbQ/W9vN7sKyFLnVQMaYRKEUb11v1Zq25yPCo=" } }, { "type": "step", "result": "=\\frac{1}{9+9u^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7smGrEegwE0OJU4NncVSnqb6tvzdVhLE832UTaXKHEMMJQJZuTAY5js+oqjdT8kslJifb1MQ4vEkJVDIbsq/4mtal+4L/VXyI0J33f2B9Oaz//NvXaLneG2moeSe54R9ow/RdK+1HAzJpFIjdMYYONTn2CKRNJ8L3Lc7WnBEoVdOwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=3\\cdot\\:\\frac{1}{9u^{2}+9}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:3}{9+9u^{2}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3=3$$", "result": "=\\frac{3}{9+9u^{2}}" }, { "type": "interim", "title": "Factor $$9+9u^{2}:{\\quad}9\\left(1+u^{2}\\right)$$", "input": "9+9u^{2}", "result": "=\\frac{3}{9\\left(1+u^{2}\\right)}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=9\\cdot\\:1+9u^{2}" }, { "type": "step", "primary": "Factor out common term $$9$$", "result": "=9\\left(1+u^{2}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=\\frac{1}{3\\left(u^{2}+1\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{3\\left(u^{2}+1\\right)}du" } ], "meta": { "interimType": "Integral Substitution 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{3\\left(u^{2}+1\\right)}du" }, { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=\\frac{1}{3}\\cdot\\:\\int\\:\\frac{1}{u^{2}+1}du" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{u^{2}+1}du=\\arctan\\left(u\\right)$$", "result": "=\\frac{1}{3}\\arctan\\left(u\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\frac{x}{3}$$", "result": "=\\frac{1}{3}\\arctan\\left(\\frac{x}{3}\\right)" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\frac{1}{3}\\arctan\\left(\\frac{x}{3}\\right)+c_{1}", "meta": { "title": { "extension": "If $$\\frac{dF\\left(x\\right)}{dx}=f\\left(x\\right)$$ then $$\\int{f\\left(x\\right)}dx=F\\left(x\\right)+C$$" } } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:-\\frac{1}{y^{2}+4}dy=-\\frac{1}{2}\\arctan\\left(\\frac{y}{2}\\right)+c_{2}$$", "input": "\\int\\:-\\frac{1}{y^{2}+4}dy", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=-\\int\\:\\frac{1}{y^{2}+4}dy" }, { "type": "interim", "title": "Apply Integral Substitution", "input": "\\int\\:\\frac{1}{y^{2}+4}dy", "steps": [ { "type": "definition", "title": "Integral Substitution definition", "text": "$$\\int\\:f\\left(g\\left(x\\right)\\right)\\cdot\\:g'\\left(x\\right)dx=\\int\\:f\\left(u\\right)du,\\:\\quad\\:u=g\\left(x\\right)$$", "secondary": [ "Substitute: $$y=2u$$" ] }, { "type": "step", "primary": "For $$by\\left(x\\right)^2\\pm\\:a\\:$$substitute $$y=\\frac{\\sqrt{a}}{\\sqrt{b}}u$$<br/>$$a=4,\\:b=1,\\:\\frac{\\sqrt{a}}{\\sqrt{b}}=2\\quad\\Rightarrow\\quad$$substitute $$y=2u$$" }, { "type": "interim", "title": "$$\\frac{dy}{du}=2$$", "input": "\\left(2u\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2u^{^{\\prime}}" }, { "type": "step", "primary": "Apply the common derivative: $$u^{\\prime}=1$$", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XyHS+/HWDjUlumUiU0YSJcPQlVFV646ejpUuEWqujX2QuIxj9K+Upo9l4tAcwN/gSLdINoPD2MyLmUXT+YnlMqNMOVZNYY3ywiaaLIKBKky+o6uaffLAsvkElaEOABE8" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dy=2du$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{\\left(2u\\right)^{2}+4}\\cdot\\:2du" }, { "type": "interim", "title": "Simplify $$\\frac{1}{\\left(2u\\right)^{2}+4}\\cdot\\:2:{\\quad}\\frac{1}{2\\left(u^{2}+1\\right)}$$", "input": "\\frac{1}{\\left(2u\\right)^{2}+4}\\cdot\\:2", "steps": [ { "type": "interim", "title": "$$\\frac{1}{\\left(2u\\right)^{2}+4}=\\frac{1}{4u^{2}+4}$$", "input": "\\frac{1}{\\left(2u\\right)^{2}+4}", "steps": [ { "type": "interim", "title": "$$\\left(2u\\right)^{2}=4u^{2}$$", "input": "\\left(2u\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "result": "=2^{2}u^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=4u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cQDj2fG3PKYzjVjILAzgjs0ag8T1MwTer44+aCS/ZFAJ/+v6bFh9qlRZ7670glkkoX4usPSd1JXjYSuz11RWtEjZIe5ncy0wi4e8qtMXCVM=" } }, { "type": "step", "result": "=\\frac{1}{4u^{2}+4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7V//FfWP5bHpmN0ZJjNhLm7Id5ggfYZeNZakyIKnsGhEJQJZuTAY5js+oqjdT8kslxJMA0gN5f5sXGhebvL4CZWoLOidZEhYwKRabIIgH71T//NvXaLneG2moeSe54R9oqLZQf25F6ZamCF/fUdTJx9oTUpB/KOSLMcsLoLDNex6wiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=2\\cdot\\:\\frac{1}{4u^{2}+4}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{4u^{2}+4}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=\\frac{2}{4u^{2}+4}" }, { "type": "interim", "title": "Factor $$4u^{2}+4:{\\quad}4\\left(u^{2}+1\\right)$$", "input": "4u^{2}+4", "result": "=\\frac{2}{4\\left(u^{2}+1\\right)}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=4u^{2}+4\\cdot\\:1" }, { "type": "step", "primary": "Factor out common term $$4$$", "result": "=4\\left(u^{2}+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{1}{2\\left(u^{2}+1\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{2\\left(u^{2}+1\\right)}du" } ], "meta": { "interimType": "Integral Substitution 1Eq" } }, { "type": "step", "result": "=-\\int\\:\\frac{1}{2\\left(u^{2}+1\\right)}du" }, { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=-\\frac{1}{2}\\cdot\\:\\int\\:\\frac{1}{u^{2}+1}du" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{u^{2}+1}du=\\arctan\\left(u\\right)$$", "result": "=-\\frac{1}{2}\\arctan\\left(u\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\frac{y}{2}$$", "result": "=-\\frac{1}{2}\\arctan\\left(\\frac{y}{2}\\right)" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-\\frac{1}{2}\\arctan\\left(\\frac{y}{2}\\right)+c_{2}", "meta": { "title": { "extension": "If $$\\frac{dF\\left(x\\right)}{dx}=f\\left(x\\right)$$ then $$\\int{f\\left(x\\right)}dx=F\\left(x\\right)+C$$" } } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "-\\frac{1}{2}\\arctan\\left(\\frac{y}{2}\\right)+c_{2}=\\frac{1}{3}\\arctan\\left(\\frac{x}{3}\\right)+c_{1}" }, { "type": "step", "primary": "Combine the constants", "result": "-\\frac{1}{2}\\arctan\\left(\\frac{y}{2}\\right)=\\frac{1}{3}\\arctan\\left(\\frac{x}{3}\\right)+c_{1}" } ], "meta": { "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Isolate $$y:{\\quad}y=2\\tan\\left(-\\frac{2}{3}\\arctan\\left(\\frac{x}{3}\\right)-2c_{1}\\right)$$", "input": "-\\frac{1}{2}\\arctan\\left(\\frac{y}{2}\\right)=\\frac{1}{3}\\arctan\\left(\\frac{x}{3}\\right)+c_{1}", "steps": [ { "type": "interim", "title": "Multiply both sides by $$-2$$", "input": "-\\frac{1}{2}\\arctan\\left(\\frac{y}{2}\\right)=\\frac{1}{3}\\arctan\\left(\\frac{x}{3}\\right)+c_{1}", "result": "\\arctan\\left(\\frac{y}{2}\\right)=-\\frac{2}{3}\\arctan\\left(\\frac{x}{3}\\right)-2c_{1}", "steps": [ { "type": "step", "primary": "Multiply both sides by $$-2$$", "result": "\\left(-\\frac{1}{2}\\arctan\\left(\\frac{y}{2}\\right)\\right)\\left(-2\\right)=\\frac{1}{3}\\arctan\\left(\\frac{x}{3}\\right)\\left(-2\\right)+c_{1}\\left(-2\\right)" }, { "type": "interim", "title": "Simplify", "input": "\\left(-\\frac{1}{2}\\arctan\\left(\\frac{y}{2}\\right)\\right)\\left(-2\\right)=\\frac{1}{3}\\arctan\\left(\\frac{x}{3}\\right)\\left(-2\\right)+c_{1}\\left(-2\\right)", "result": "\\arctan\\left(\\frac{y}{2}\\right)=-\\frac{2}{3}\\arctan\\left(\\frac{x}{3}\\right)-2c_{1}", "steps": [ { "type": "interim", "title": "Simplify $$\\left(-\\frac{1}{2}\\arctan\\left(\\frac{y}{2}\\right)\\right)\\left(-2\\right):{\\quad}\\arctan\\left(\\frac{y}{2}\\right)$$", "input": "\\left(-\\frac{1}{2}\\arctan\\left(\\frac{y}{2}\\right)\\right)\\left(-2\\right)", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{1}{2}\\arctan\\left(\\frac{y}{2}\\right)\\cdot\\:2" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}\\arctan\\left(\\frac{y}{2}\\right)" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\arctan\\left(\\frac{y}{2}\\right)\\cdot\\:1" }, { "type": "step", "primary": "Multiply: $$\\arctan\\left(\\frac{y}{2}\\right)\\cdot\\:1=\\arctan\\left(\\frac{y}{2}\\right)$$", "result": "=\\arctan\\left(\\frac{y}{2}\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DPqTQEO9IJ0HThO5ILSNFy1LUpF0x/Pk9w0OvzQSes+8Z2cNhR8DK7EPmxWye6XV3XeO2tIUPH5Q2xrCOU6NXaLuQQnwsptiBQ95NT8ooYNiwU48LMb5Gwj43Kmq8eRSo3oe/oyhMy2+1TQhDBd2f2zM6E3fuZxF1XkKAYaRXCBPUT0anQaTGZRSu0p4syd+hqoR7vBuiov198ixsaLGytPH7HRxTTatuzF7OlCEegQ=" } }, { "type": "interim", "title": "Simplify $$\\frac{1}{3}\\arctan\\left(\\frac{x}{3}\\right)\\left(-2\\right)+c_{1}\\left(-2\\right):{\\quad}-\\frac{2}{3}\\arctan\\left(\\frac{x}{3}\\right)-2c_{1}$$", "input": "\\frac{1}{3}\\arctan\\left(\\frac{x}{3}\\right)\\left(-2\\right)+c_{1}\\left(-2\\right)", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\frac{1}{3}\\arctan\\left(\\frac{x}{3}\\right)\\cdot\\:2-c_{1}\\cdot\\:2" }, { "type": "interim", "title": "$$\\frac{1}{3}\\arctan\\left(\\frac{x}{3}\\right)\\cdot\\:2=\\frac{2}{3}\\arctan\\left(\\frac{x}{3}\\right)$$", "input": "\\frac{1}{3}\\arctan\\left(\\frac{x}{3}\\right)\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{3}\\arctan\\left(\\frac{x}{3}\\right)" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=\\frac{2}{3}\\arctan\\left(\\frac{x}{3}\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71Hcipo16ybrRdbQSFyXFL/qo5EFjx5oE6+wUVutvqGIxrsMB0uR3vkXBH25xyw0ZICf2WQN9mSJxQaQ7cQX4il2BgpogRffZXrJl2fYiHyfyOI2a95ZWD7O1GbogqjphhOndYV+p9tE1n9dVbLTLpA4bfwiV6iMLJ5sC1nL7dOZUgeqPvqS0abqc3bZlmX78YWW+ks1bjo4ncxE4ipuwtJsN7Jx4V0G0x7taOuLPausgU1KmKXIcjqtZvwDnOztpmjdJ5babVt29i4JyP34NhA==" } }, { "type": "step", "result": "=-\\frac{2}{3}\\arctan\\left(\\frac{x}{3}\\right)-2c_{1}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71Hcipo16ybrRdbQSFyXFL/qo5EFjx5oE6+wUVutvqGI/qTLTG7lSgAWfmR3rYLv/AsQc8Xk3/5KRmZeYKWN7RHm0yklMEFxxRaRLK8x7VKnlaD0HMlhDXFpQoVp2Ux1TKSrQX49kAIjZnrigexSEFglwr159ATyhM0blK/8VnrDWwPs1+Gw97t4MeuaNjSYTwPBBw3byB5RPtpSy6YLt2rjIGkp2RTIQvGlCN+Yt76I9SaDJcDI6c6XxjDG+uVczNecgkw3yvMkGcOWM2TD6OA==" } }, { "type": "step", "result": "\\arctan\\left(\\frac{y}{2}\\right)=-\\frac{2}{3}\\arctan\\left(\\frac{x}{3}\\right)-2c_{1}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Apply trig inverse properties", "input": "\\arctan\\left(\\frac{y}{2}\\right)=-\\frac{2}{3}\\arctan\\left(\\frac{x}{3}\\right)-2c_{1}", "result": "\\frac{y}{2}=\\tan\\left(-\\frac{2}{3}\\arctan\\left(\\frac{x}{3}\\right)-2c_{1}\\right)", "steps": [ { "type": "step", "primary": "$$\\arctan\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\tan\\left(a\\right)$$", "result": "\\frac{y}{2}=\\tan\\left(-\\frac{2}{3}\\arctan\\left(\\frac{x}{3}\\right)-2c_{1}\\right)" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } }, { "type": "interim", "title": "Solve $$\\frac{y}{2}=\\tan\\left(-\\frac{2}{3}\\arctan\\left(\\frac{x}{3}\\right)-2c_{1}\\right):{\\quad}y=2\\tan\\left(-\\frac{2}{3}\\arctan\\left(\\frac{x}{3}\\right)-2c_{1}\\right)$$", "input": "\\frac{y}{2}=\\tan\\left(-\\frac{2}{3}\\arctan\\left(\\frac{x}{3}\\right)-2c_{1}\\right)", "steps": [ { "type": "interim", "title": "Multiply both sides by $$2$$", "input": "\\frac{y}{2}=\\tan\\left(-\\frac{2}{3}\\arctan\\left(\\frac{x}{3}\\right)-2c_{1}\\right)", "result": "y=2\\tan\\left(-\\frac{2}{3}\\arctan\\left(\\frac{x}{3}\\right)-2c_{1}\\right)", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{2y}{2}=2\\tan\\left(-\\frac{2}{3}\\arctan\\left(\\frac{x}{3}\\right)-2c_{1}\\right)" }, { "type": "step", "primary": "Simplify", "result": "y=2\\tan\\left(-\\frac{2}{3}\\arctan\\left(\\frac{x}{3}\\right)-2c_{1}\\right)" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "y=2\\tan\\left(-\\frac{2}{3}\\arctan\\left(\\frac{x}{3}\\right)-2c_{1}\\right)" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Generic Isolate Title 1Eq" } }, { "type": "step", "primary": "Simplify", "result": "y=2\\tan\\left(-\\frac{2}{3}\\arctan\\left(\\frac{x}{3}\\right)-c_{1}\\right)" } ], "meta": { "interimType": "ODE Solve Separable 0Eq" } }, { "type": "step", "result": "y=2\\tan\\left(-\\frac{2}{3}\\arctan\\left(\\frac{x}{3}\\right)-c_{1}\\right)" } ], "meta": { "solvingClass": "ODE" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "#>#ODE#>#y=2\\tan(-\\frac{2}{3}\\arctan(\\frac{x}{3})-c_{1})" } } }, "meta": { "showVerify": true } }