4xy^'-20y=x^{-5}

{ "query": { "display": "$$4xy^{^{\\prime}}-20y=x^{-5}$$", "symbolab_question": "ODE#4xy^{\\prime }-20y=x^{-5}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "ODE", "subTopic": "FirstLinear", "default": "y=-\\frac{1}{40x^{5}}+c_{1}x^{5}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$4xy^{\\prime}\\left(x\\right)-20y=x^{-5}:{\\quad}y=-\\frac{1}{40x^{5}}+c_{1}x^{5}$$", "input": "4xy^{\\prime}\\left(x\\right)-20y=x^{-5}", "steps": [ { "type": "interim", "title": "Solve linear ODE:$${\\quad}y=-\\frac{1}{40x^{5}}+c_{1}x^{5}$$", "input": "4xy^{\\prime}\\left(x\\right)-20y=x^{-5}", "steps": [ { "type": "definition", "title": "First order linear Ordinary Differential Equation", "text": "A first order linear ODE has the form of $$y'\\left(x\\right)+p\\left(x\\right)y=q\\left(x\\right)$$" }, { "type": "interim", "title": "Rewrite in the form of a first order linear ODE", "input": "4xy^{\\prime}\\left(x\\right)-20y=x^{-5}", "result": "y^{\\prime}\\left(x\\right)-\\frac{5}{x}y=\\frac{1}{4x^{6}}", "steps": [ { "type": "step", "primary": "Standard form of a first order linear ODE:", "secondary": [ "$$y'\\left(x\\right)+p\\left(x\\right){\\cdot}y=q\\left(x\\right)$$" ] }, { "type": "step", "result": "4xy^{^{\\prime}}\\left(x\\right)-20y=x^{-5}" }, { "type": "step", "primary": "Simplify", "result": "4xy^{^{\\prime}}\\left(x\\right)-20y=\\frac{1}{x^{5}}" }, { "type": "step", "primary": "Divide both sides by $$4x$$", "result": "\\frac{4xy^{^{\\prime}}\\left(x\\right)}{4x}-\\frac{20y}{4x}=\\frac{\\frac{1}{x^{5}}}{4x}" }, { "type": "step", "primary": "Simplify", "result": "y^{^{\\prime}}\\left(x\\right)-\\frac{5y}{x}=\\frac{1}{4x^{6}}" }, { "type": "step", "primary": "Rewrite in standard form", "secondary": [ "$$p\\left(x\\right)=-\\frac{5}{x},\\:{\\quad}q\\left(x\\right)=\\frac{1}{4x^{6}}$$" ], "result": "y^{^{\\prime}}\\left(x\\right)-\\frac{5}{x}y=\\frac{1}{4x^{6}}" } ], "meta": { "interimType": "Canon First Order ODE 2Eq" } }, { "type": "interim", "title": "Find the integration factor:$${\\quad}μ\\left(x\\right)=\\frac{1}{x^{5}}$$", "steps": [ { "type": "step", "primary": "Find the integrating factor $$\\mu\\left(x\\right)$$, so that: $$\\mu\\left(x\\right){\\cdot}p\\left(x\\right)=\\mu'\\left(x\\right)$$", "result": "μ^{^{\\prime}}\\left(x\\right)=μ\\left(x\\right)p\\left(x\\right)" }, { "type": "step", "primary": "Divide both sides by $$μ\\left(x\\right)$$", "result": "\\frac{μ^{^{\\prime}}\\left(x\\right)}{μ\\left(x\\right)}=\\frac{μ\\left(x\\right)p\\left(x\\right)}{μ\\left(x\\right)}" }, { "type": "step", "primary": "Simplify", "result": "\\frac{μ^{^{\\prime}}\\left(x\\right)}{μ\\left(x\\right)}=p\\left(x\\right)" }, { "type": "step", "primary": "$$\\left(\\ln\\left(μ\\left(x\\right)\\right)\\right)^{\\prime}=\\frac{μ^{\\prime}\\left(x\\right)}{μ\\left(x\\right)}$$", "result": "\\left(\\ln\\left(μ\\left(x\\right)\\right)\\right)^{^{\\prime}}=p\\left(x\\right)", "meta": { "general_rule": { "extension": "$$\\frac{d}{dx}\\left(\\ln\\left(f\\left(x\\right)\\right)\\right)=\\frac{\\frac{d}{dx}f\\left(x\\right)}{f\\left(x\\right)}$$" } } }, { "type": "step", "primary": "$$p\\left(x\\right)=-\\frac{5}{x}$$", "result": "\\left(\\ln\\left(μ\\left(x\\right)\\right)\\right)^{^{\\prime}}=-\\frac{5}{x}" }, { "type": "interim", "title": "Solve $$\\left(\\ln\\left(μ\\left(x\\right)\\right)\\right)^{\\prime}=-\\frac{5}{x}:{\\quad}μ\\left(x\\right)=\\frac{e^{c_{1}}}{x^{5}}$$", "input": "\\left(\\ln\\left(μ\\left(x\\right)\\right)\\right)^{\\prime}=-\\frac{5}{x}", "steps": [ { "type": "step", "primary": "If$${\\quad}f^{^{\\prime}}\\left(x\\right)=g\\left(x\\right){\\quad}$$then$${\\quad}f\\left(x\\right)=\\int{g\\left(x\\right)}dx$$", "result": "\\ln\\left(μ\\left(x\\right)\\right)=\\int\\:-\\frac{5}{x}dx" }, { "type": "interim", "title": "$$\\int\\:-\\frac{5}{x}dx=-5\\ln\\left(x\\right)+c_{1}$$", "input": "\\int\\:-\\frac{5}{x}dx", "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": "=-5\\cdot\\:\\int\\:\\frac{1}{x}dx" }, { "type": "step", "primary": "Use the common integral:$$\\quad\\:\\int\\:\\frac{1}{x}dx=\\ln\\left(x\\right),\\:$$assuming a complex-valued logarithm", "result": "=-5\\ln\\left(x\\right)" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-5\\ln\\left(x\\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": "step", "result": "\\ln\\left(μ\\left(x\\right)\\right)=-5\\ln\\left(x\\right)+c_{1}" }, { "type": "interim", "title": "Isolate $$μ\\left(x\\right):{\\quad}μ\\left(x\\right)=\\frac{e^{c_{1}}}{x^{5}}$$", "input": "\\ln\\left(μ\\left(x\\right)\\right)=-5\\ln\\left(x\\right)+c_{1}", "steps": [ { "type": "interim", "title": "Apply log rules", "input": "\\ln\\left(μ\\left(x\\right)\\right)=-5\\ln\\left(x\\right)+c_{1}", "result": "μ\\left(x\\right)=\\frac{e^{c_{1}}}{x^{5}}", "steps": [ { "type": "step", "primary": "Use the logarithmic definition: If $$\\log_a\\left(b\\right)=c\\:$$then $$b=a^c$$", "secondary": [ "$$\\ln\\left(μ\\left(x\\right)\\right)=-5\\ln\\left(x\\right)+c_{1}\\quad\\:\\Rightarrow\\:\\quad\\:μ\\left(x\\right)=e^{-5\\ln\\left(x\\right)+c_{1}}$$" ], "result": "μ\\left(x\\right)=e^{-5\\ln\\left(x\\right)+c_{1}}" }, { "type": "interim", "title": "Expand $$e^{-5\\ln\\left(x\\right)+c_{1}}:{\\quad}\\frac{e^{c_{1}}}{x^{5}}$$", "input": "e^{-5\\ln\\left(x\\right)+c_{1}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "result": "=e^{-5\\ln\\left(x\\right)}e^{c_{1}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "Simplify $$e^{-5\\ln\\left(x\\right)}:{\\quad}x^{-5}$$", "input": "e^{-5\\ln\\left(x\\right)}", "result": "=x^{-5}e^{c_{1}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{bc}=\\left(a^{b}\\right)^{c}$$", "result": "=\\left(e^{\\ln\\left(x\\right)}\\right)^{-5}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply log rule: $$a^{\\log_{a}\\left(b\\right)}=b$$", "secondary": [ "$$e^{\\ln\\left(x\\right)}=x$$" ], "result": "=x^{-5}", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "Simplify", "input": "x^{-5}e^{c_{1}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$x^{-5}=\\frac{1}{x^{5}}$$" ], "result": "=e^{c_{1}}\\frac{1}{x^{5}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:e^{c_{1}}}{x^{5}}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:e^{c_{1}}=e^{c_{1}}$$", "result": "=\\frac{e^{c_{1}}}{x^{5}}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } }, { "type": "step", "result": "=\\frac{e^{c_{1}}}{x^{5}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sWYySz6F4K+z3jy8SMIUx1SMalNvsAeBNmfxNaYrZwO+sC8ksde3EL3dW7N7EScc6upJ8tRrcqdzpqw4uWy22Tu/0m71n2FfPReH2RI2dnj8bYA0b6V2RSTOZ7Os9NODUQnGeUzq9noEBkFKC8/pLNQPMmtpIekbY9TH2qp6FCs=" } }, { "type": "step", "result": "μ\\left(x\\right)=\\frac{e^{c_{1}}}{x^{5}}" } ], "meta": { "interimType": "Apply Log Rules Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xZ/kvw9iKo4JlZMns/8rf1NQeHdZDE8/hcXcm5x7D2YmSzwWcMgxhWR3+PQclu59hqX8q7xCWwsRcJY5APRK6siNG73ekXpR3ysC7LzMiyY2BRsOoi1F45bY11mCIgwm/4geG+870335qskweTP8M4yg3Y6Vz5Vyt8tIdLH1viQEuDOVaQvKofqHoY5jNapsVI7xSbweWtYxw/Tkw5qK4g==" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Isolate Title 1Eq" } }, { "type": "step", "result": "μ\\left(x\\right)=\\frac{e^{c_{1}}}{x^{5}}" } ], "meta": { "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "μ\\left(x\\right)=\\frac{e^{c_{1}}}{x^{5}}" }, { "type": "step", "primary": "The constant $$e^{c_{1}}\\:$$can be dropped (it will be absorbed into C)", "result": "μ\\left(x\\right)=\\frac{1}{x^{5}}" } ], "meta": { "interimType": "Integrating Factor Top 0Eq" } }, { "type": "interim", "title": "Put the equation in the form $$\\left(\\mu\\left(x\\right){\\cdot}y\\right)'=\\mu\\left(x\\right){\\cdot}q\\left(x\\right):{\\quad}\\left(\\frac{1}{x^{5}}y\\right)^{\\prime}=\\frac{1}{4x^{11}}$$", "steps": [ { "type": "step", "primary": "Multiply by the integration factor, $$\\mu\\left(x\\right)$$ and rewrtie the equation as<br/>$$\\left(\\mu\\left(x\\right)\\cdot\\:y\\left(x\\right)\\right)'=\\mu\\:\\left(x\\right)\\cdot\\:q\\left(x\\right)$$", "result": "y^{^{\\prime}}\\left(x\\right)-\\frac{5}{x}y=\\frac{1}{4x^{6}}" }, { "type": "step", "primary": "Multiply both sides by the integrating factor, $$\\frac{1}{x^{5}}$$", "result": "y^{^{\\prime}}\\left(x\\right)\\frac{1}{x^{5}}-\\frac{5}{x}y\\frac{1}{x^{5}}=\\frac{1\\cdot\\:\\frac{1}{x^{5}}}{4x^{6}}" }, { "type": "step", "primary": "Simplify", "result": "\\frac{y^{^{\\prime}}\\left(x\\right)}{x^{5}}-\\frac{5y}{x^{6}}=\\frac{1}{4x^{11}}" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\frac{1}{x^{5}},\\:g=y:{\\quad}\\frac{y^{\\prime}\\left(x\\right)}{x^{5}}-\\frac{5y}{x^{6}}=\\left(\\frac{1}{x^{5}}y\\right)^{\\prime}$$" ], "result": "\\left(\\frac{1}{x^{5}}y\\right)^{^{\\prime}}=\\frac{1}{4x^{11}}" } ], "meta": { "interimType": "Bring Linear To Derivative Form Left 0Eq" } }, { "type": "interim", "title": "Solve $$\\left(\\frac{1}{x^{5}}y\\right)^{\\prime}=\\frac{1}{4x^{11}}:{\\quad}y=-\\frac{1}{40x^{5}}+c_{1}x^{5}$$", "input": "\\left(\\frac{1}{x^{5}}y\\right)^{\\prime}=\\frac{1}{4x^{11}}", "steps": [ { "type": "step", "primary": "If$${\\quad}f^{^{\\prime}}\\left(x\\right)=g\\left(x\\right){\\quad}$$then$${\\quad}f\\left(x\\right)=\\int{g\\left(x\\right)}dx$$", "result": "\\frac{1}{x^{5}}y=\\int\\:\\frac{1}{4x^{11}}dx" }, { "type": "interim", "title": "$$\\int\\:\\frac{1}{4x^{11}}dx=-\\frac{1}{40x^{10}}+c_{1}$$", "input": "\\int\\:\\frac{1}{4x^{11}}dx", "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": "=\\frac{1}{4}\\cdot\\:\\int\\:\\frac{1}{x^{11}}dx" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:\\frac{1}{x^{11}}dx", "result": "=\\frac{1}{4}\\left(-\\frac{1}{10x^{10}}\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a^b}=a^{-b}$$", "secondary": [ "$$\\frac{1}{x^{11}}=x^{-11}$$" ], "result": "=\\int\\:x^{-11}dx", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{x^{-11+1}}{-11+1}" }, { "type": "interim", "title": "Simplify $$\\frac{x^{-11+1}}{-11+1}:{\\quad}-\\frac{1}{10x^{10}}$$", "input": "\\frac{x^{-11+1}}{-11+1}", "steps": [ { "type": "step", "primary": "Add/Subtract the numbers: $$-11+1=-10$$", "result": "=\\frac{x^{-10}}{-10}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{x^{-10}}{10}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$x^{-10}=\\frac{1}{x^{10}}$$" ], "result": "=-\\frac{\\frac{1}{x^{10}}}{10}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "secondary": [ "$$\\frac{\\frac{1}{x^{10}}}{10}=\\frac{1}{x^{10}\\cdot\\:10}$$" ], "result": "=-\\frac{1}{x^{10}\\cdot\\:10}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=-\\frac{1}{10x^{10}}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7/nALbhiEHRW5Du1KueZP+r6gQW8jfeVgl4gkdRj7fNS/ycil7jvdDYGLhLUZ1RVKEBIE8bEA3G5Au5Z7P1qm6TRcFZ1kk1FeGgy6pS6cJVMm7IyZlDD6ihahH730JdDhoEFMST8lDZxn1Yq5HMKVTsN/SZgQjH1OoahVjOHG2Hrj8LNU2fafRgGTDrnDOEnog==" } }, { "type": "interim", "title": "Simplify $$\\frac{1}{4}\\left(-\\frac{1}{10x^{10}}\\right):{\\quad}-\\frac{1}{40x^{10}}$$", "input": "\\frac{1}{4}\\left(-\\frac{1}{10x^{10}}\\right)", "result": "=-\\frac{1}{40x^{10}}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\frac{1}{4}\\cdot\\:\\frac{1}{10x^{10}}" }, { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=-\\frac{1\\cdot\\:1}{4\\cdot\\:10x^{10}}" }, { "type": "step", "primary": "Refine", "result": "=-\\frac{1}{40x^{10}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73KsJ3XTwG6XEOk77cUS7DyG8BV/AR8CMKUJaKsVPs493/Cc9UTtlx0dAKwOq7SolzMFYmi1F5Hg/ibpEToVnYzUT/nWcGX2Ursa9ihRU/ogWTtq7yydaXO10JWOuXWnDixpjZyz1hn/Tx95aZo6CSFjbCAuoanhwjo/9bMp/72l6XrZsV/ESKikalXMjGq5d+qxG6RKk8sqt4+5+benR2w==" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-\\frac{1}{40x^{10}}+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": "step", "result": "\\frac{1}{x^{5}}y=-\\frac{1}{40x^{10}}+c_{1}" }, { "type": "interim", "title": "Isolate $$y:{\\quad}y=-\\frac{1}{40x^{5}}+c_{1}x^{5}$$", "input": "\\frac{1}{x^{5}}y=-\\frac{1}{40x^{10}}+c_{1}", "steps": [ { "type": "interim", "title": "Multiply both sides by $$x^{5}$$", "input": "\\frac{1}{x^{5}}y=-\\frac{1}{40x^{10}}+c_{1}", "result": "y=-\\frac{1}{40x^{5}}+c_{1}x^{5}", "steps": [ { "type": "step", "primary": "Multiply both sides by $$x^{5}$$", "result": "\\frac{1}{x^{5}}yx^{5}=-\\frac{1}{40x^{10}}x^{5}+c_{1}x^{5}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{1}{x^{5}}yx^{5}=-\\frac{1}{40x^{10}}x^{5}+c_{1}x^{5}", "result": "y=-\\frac{1}{40x^{5}}+c_{1}x^{5}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{1}{x^{5}}yx^{5}:{\\quad}y$$", "input": "\\frac{1}{x^{5}}yx^{5}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:yx^{5}}{x^{5}}" }, { "type": "step", "primary": "Cancel the common factor: $$x^{5}$$", "result": "=1\\cdot\\:y" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:y=y$$", "result": "=y" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7LKMAzu53FlM68o+ufxLuo6bQYYE70V5p6QbwyxvXGmN1g99dC9fj9sg0EHzBIRDRAYB3vtqewg74cpbGyz0PsR429vuTSxWa7B/X3D1oP03AWQmX+FAZQ57eQ8HwbCJCK8m84Elhvot5yA3cm9kkyr8yD3hLQ33B7/8/LpbPE3o=" } }, { "type": "interim", "title": "Simplify $$-\\frac{1}{40x^{10}}x^{5}+c_{1}x^{5}:{\\quad}-\\frac{1}{40x^{5}}+c_{1}x^{5}$$", "input": "-\\frac{1}{40x^{10}}x^{5}+c_{1}x^{5}", "steps": [ { "type": "interim", "title": "$$\\frac{1}{40x^{10}}x^{5}=\\frac{1}{40x^{5}}$$", "input": "\\frac{1}{40x^{10}}x^{5}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:x^{5}}{40x^{10}}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:x^{5}=x^{5}$$", "result": "=\\frac{x^{5}}{40x^{10}}" }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=\\frac{1}{x^{b-a}}$$", "secondary": [ "$$\\frac{x^{5}}{x^{10}}=\\frac{1}{x^{10-5}}$$" ], "result": "=\\frac{1}{40x^{10-5}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$10-5=5$$", "result": "=\\frac{1}{40x^{5}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Mj2FdotmvNXijuuxp1h+vylwQ/fV846f2Vng826QrtjNGoPE9TME3q+OPmgkv2RQiEw6G4T+RFI2ZfZDoB3kMqWyRiKo1JVvfU/zRhmiqepTW26qciuyUBGXQExCUedYrrIvUHFUcnszR9pYM6BHXTFsccslvr8047AmUaoOwYJzSxziqikvT7jRuqiMEU90" } }, { "type": "step", "result": "=-\\frac{1}{40x^{5}}+c_{1}x^{5}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xT/bFXl2GdKJHJQ3njlEMCHw+aCblH/GmkykGWSJb1lCVuHA46gBQAvtvkV/+Fy1q47vuWedXv2WUg94ER8IwboFh68UBPsKD7+y2r59jcTG9Kzs/4Hn+IWEJZ+9KSNm/z//r+dXk7h9vxeDCLuZqnKF3u2OIb4bFA3EO8aRlSWC7ONLOje88Peuuzx1JFjGTZcwXW2C8dZ8Jz9RJ+IHWTVz4TvKfHsTeOu1LEL4Awc=" } }, { "type": "step", "result": "y=-\\frac{1}{40x^{5}}+c_{1}x^{5}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": 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"y=-\\frac{1}{40x^{5}}+c_{1}x^{5}" } ], "meta": { "solvingClass": "ODE" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "#>#ODE#>#y=-\\frac{1}{40x^{5}}+c_{1}x^{5}" } } }, "meta": { "showVerify": true } }