y^'=(x^2y-y)/(y+1)

{ "query": { "display": "$$y^{^{\\prime}}=\\frac{x^{2}y-y}{y+1}$$", "symbolab_question": "ODE#y^{\\prime }=\\frac{x^{2}y-y}{y+1}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "ODE", "subTopic": "FirstSeparable", "default": "y=\\W(\\frac{1}{e^{\\frac{-x^{3}+3x-c_{1}}{3}}})", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$y^{\\prime}\\left(x\\right)=\\frac{x^{2}y-y}{y+1}:{\\quad}y=\\W\\left(\\frac{1}{e^{\\frac{-x^{3}+3x-c_{1}}{3}}}\\right)$$", "input": "y^{\\prime}\\left(x\\right)=\\frac{x^{2}y-y}{y+1}", "steps": [ { "type": "interim", "title": "Solve separable ODE:$${\\quad}y=\\W\\left(\\frac{1}{e^{\\frac{-x^{3}+3x-c_{1}}{3}}}\\right)$$", "input": "y^{\\prime}\\left(x\\right)=\\frac{x^{2}y-y}{y+1}", "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": "interim", "title": "Rewrite in the form of a first order separable ODE", "input": "y^{\\prime}\\left(x\\right)=\\frac{x^{2}y-y}{y+1}", "result": "\\frac{y+1}{y}y^{\\prime}\\left(x\\right)=x^{2}-1", "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": "y^{^{\\prime}}\\left(x\\right)=\\frac{x^{2}y-y}{y+1}" }, { "type": "step", "primary": "Multiply both sides by $$y+1$$", "result": "y^{^{\\prime}}\\left(x\\right)\\left(y+1\\right)=\\frac{x^{2}y-y}{y+1}\\left(y+1\\right)" }, { "type": "step", "primary": "Simplify", "result": "y^{^{\\prime}}\\left(x\\right)\\left(y+1\\right)=x^{2}y-y" }, { "type": "step", "primary": "Divide both sides by $$y$$", "result": "\\frac{y^{^{\\prime}}\\left(x\\right)\\left(y+1\\right)}{y}=\\frac{x^{2}y-y}{y}" }, { "type": "step", "primary": "Simplify", "result": "\\frac{y^{^{\\prime}}\\left(x\\right)\\left(y+1\\right)}{y}=x^{2}-1" }, { "type": "step", "primary": "Rewrite in standard form", "secondary": [ "$$N\\left(y\\right)=\\frac{y+1}{y},\\:{\\quad}M\\left(x\\right)=x^{2}-1$$" ], "result": "\\frac{y+1}{y}y^{^{\\prime}}\\left(x\\right)=x^{2}-1" } ], "meta": { "interimType": "Canon First Order Separable ODE 2Eq" } }, { "type": "interim", "title": "Solve $$\\frac{y+1}{y}y^{\\prime}\\left(x\\right)=x^{2}-1:{\\quad}y+\\ln\\left(y\\right)=\\frac{x^{3}}{3}-x+c_{1}$$", "input": "\\frac{y+1}{y}y^{\\prime}\\left(x\\right)=x^{2}-1", "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{y+1}{y}dy=\\int\\:x^{2}-1dx" }, { "type": "step", "primary": "Integrate each side of the equation" }, { "type": "interim", "title": "$$\\int\\:x^{2}-1dx=\\frac{x^{3}}{3}-x+c_{1}$$", "input": "\\int\\:x^{2}-1dx", "steps": [ { "type": "step", "primary": "Apply the Sum Rule: $$\\int{f\\left(x\\right){\\pm}g\\left(x\\right)}dx=\\int{f\\left(x\\right)}dx{\\pm}\\int{g\\left(x\\right)}dx$$", "result": "=\\int\\:x^{2}dx-\\int\\:1dx" }, { "type": "interim", "title": "$$\\int\\:x^{2}dx=\\frac{x^{3}}{3}$$", "input": "\\int\\:x^{2}dx", "steps": [ { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:x^{2}dx", "result": "=\\frac{x^{3}}{3}", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{x^{2+1}}{2+1}" }, { "type": "interim", "title": "Simplify $$\\frac{x^{2+1}}{2+1}:{\\quad}\\frac{x^{3}}{3}$$", "input": "\\frac{x^{2+1}}{2+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=\\frac{x^{3}}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{x^{3}}{3}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7+w+ikB2VyJnNfLrQuoxvVyo/JI5bBgpgExN510TA5cyodqSCYnUP+KiNK7E2zlYiE/QYMzREewyYhmRoDar7odVISTIak7VD9OG2tlObqsigQUxJPyUNnGfVirkcwpVOw39JmBCMfU6hqFWM4cbYeuPws1TZ9p9GAZMOucM4Sei" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:1dx=x$$", "input": "\\int\\:1dx", "steps": [ { "type": "step", "primary": "Integral of a constant: $$\\int{a}dx=ax$$", "result": "=1\\cdot\\:x" }, { "type": "step", "primary": "Simplify", "result": "=x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=\\frac{x^{3}}{3}-x" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\frac{x^{3}}{3}-x+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": "$$\\int\\:\\frac{y+1}{y}dy=y+\\ln\\left(y\\right)+c_{2}$$", "input": "\\int\\:\\frac{y+1}{y}dy", "steps": [ { "type": "interim", "title": "Expand $$\\frac{y+1}{y}:{\\quad}1+\\frac{1}{y}$$", "input": "\\frac{y+1}{y}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a\\pm\\:b}{c}=\\frac{a}{c}\\pm\\:\\frac{b}{c}$$", "secondary": [ "$$\\frac{y+1}{y}=\\frac{y}{y}+\\frac{1}{y}$$" ], "result": "=\\frac{y}{y}+\\frac{1}{y}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "secondary": [ "$$\\frac{y}{y}=1$$" ], "result": "=1+\\frac{1}{y}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7V9Hm/Hi/wB/Dr5csOHP1JwCWKUbvV6WK3fDUgFtg3Q+KYOk5NG6qmN1otcVUKuG36RMTM5uIqSGry95Bl21AdIEFMST8lDZxn1Yq5HMKVTvDpCb0LomfCc/BDlzuuIuGYTMwkcFqBmvA6HealqYFoQ==" } }, { "type": "step", "result": "=\\int\\:1+\\frac{1}{y}dy" }, { "type": "step", "primary": "Apply the Sum Rule: $$\\int{f\\left(x\\right){\\pm}g\\left(x\\right)}dx=\\int{f\\left(x\\right)}dx{\\pm}\\int{g\\left(x\\right)}dx$$", "result": "=\\int\\:1dy+\\int\\:\\frac{1}{y}dy" }, { "type": "interim", "title": "$$\\int\\:1dy=y$$", "input": "\\int\\:1dy", "steps": [ { "type": "step", "primary": "Integral of a constant: $$\\int{a}dx=ax$$", "result": "=1\\cdot\\:y" }, { "type": "step", "primary": "Simplify", "result": "=y", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\frac{1}{y}dy=\\ln\\left(y\\right)$$", "input": "\\int\\:\\frac{1}{y}dy", "steps": [ { "type": "step", "primary": "Use the common integral:$$\\quad\\:\\int\\:\\frac{1}{y}dy=\\ln\\left(y\\right),\\:$$assuming a complex-valued logarithm", "result": "=\\ln\\left(y\\right)" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=y+\\ln\\left(y\\right)" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=y+\\ln\\left(y\\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": "y+\\ln\\left(y\\right)+c_{2}=\\frac{x^{3}}{3}-x+c_{1}" }, { "type": "step", "primary": "Combine the constants", "result": "y+\\ln\\left(y\\right)=\\frac{x^{3}}{3}-x+c_{1}" } ], "meta": { "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Isolate $$y:{\\quad}y=\\W\\left(\\frac{1}{e^{\\frac{-x^{3}+3x-3c_{1}}{3}}}\\right)$$", "input": "y+\\ln\\left(y\\right)=\\frac{x^{3}}{3}-x+c_{1}", "steps": [ { "type": "interim", "title": "Move $$y\\:$$to the right side", "input": "y+\\ln\\left(y\\right)=\\frac{x^{3}}{3}-x+c_{1}", "result": "\\ln\\left(y\\right)=\\frac{x^{3}}{3}-x+c_{1}-y", "steps": [ { "type": "step", "primary": "Subtract $$y$$ from both sides", "result": "y+\\ln\\left(y\\right)-y=\\frac{x^{3}}{3}-x+c_{1}-y" }, { "type": "step", "primary": "Simplify", "result": "\\ln\\left(y\\right)=\\frac{x^{3}}{3}-x+c_{1}-y" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "step", "primary": "If $$f\\left(x\\right)=g\\left(x\\right),\\:$$then $$a^{f\\left(x\\right)}=a^{g\\left(x\\right)}$$", "result": "e^{\\ln\\left(y\\right)}=e^{\\frac{x^{3}}{3}-x+c_{1}-y}" }, { "type": "interim", "title": "Simplify $$e^{\\ln\\left(y\\right)}:{\\quad}y$$", "input": "e^{\\ln\\left(y\\right)}", "steps": [ { "type": "step", "primary": "Apply log rule: $$a^{\\log_{a}\\left(b\\right)}=b$$", "result": "=y", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yT8UWcrAX4HJqK9oEuELG96GQqufR6tr2vPxOUv7H++Sgn/5FOzi6lTt0R1yfVr9gQUxJPyUNnGfVirkcwpVO0SwfnDPFvkY2ovSL04Ql7b1yv0wwcIgurq/g3KjquVM" } }, { "type": "step", "result": "y=e^{\\frac{x^{3}}{3}-x+c_{1}-y}" }, { "type": "interim", "title": "Solve $$y=e^{\\frac{x^{3}}{3}-x+c_{1}-y}:{\\quad}y=\\W\\left(\\frac{1}{e^{\\frac{-x^{3}+3x-3c_{1}}{3}}}\\right)$$", "input": "y=e^{\\frac{x^{3}}{3}-x+c_{1}-y}", "result": "y=\\W\\left(\\frac{1}{e^{\\frac{-x^{3}+3x-3c_{1}}{3}}}\\right)", "steps": [ { "type": "interim", "title": "Rewrite $$y=e^{\\frac{x^{3}}{3}-x+c_{1}-y}\\:$$in Lambert form:$${\\quad}e^{y}y=\\frac{1}{e^{\\frac{-x^{3}+3x-3c_{1}}{3}}}$$", "input": "y=e^{\\frac{x^{3}}{3}-x+c_{1}-y}", "steps": [ { "type": "step", "primary": "$$xe^x=a\\:$$is equation in Lambert form" }, { "type": "step", "primary": "Multiply both sides by $$e^{-\\frac{x^{3}}{3}+x-c_{1}+y}$$", "result": "ye^{-\\frac{x^{3}}{3}+x-c_{1}+y}=e^{\\frac{x^{3}}{3}-x+c_{1}-y}e^{-\\frac{x^{3}}{3}+x-c_{1}+y}" }, { "type": "interim", "title": "Simplify $$e^{\\frac{x^{3}}{3}-x+c_{1}-y}e^{-\\frac{x^{3}}{3}+x-c_{1}+y}:{\\quad}1$$", "input": "e^{\\frac{x^{3}}{3}-x+c_{1}-y}e^{-\\frac{x^{3}}{3}+x-c_{1}+y}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$e^{\\frac{x^{3}}{3}-x+c_{1}-y}e^{-\\frac{x^{3}}{3}+x-c_{1}+y}=\\:e^{\\frac{x^{3}}{3}-x+c_{1}-y-\\frac{x^{3}}{3}+x-c_{1}+y}$$" ], "result": "=e^{\\frac{x^{3}}{3}-x+c_{1}-y-\\frac{x^{3}}{3}+x-c_{1}+y}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "Simplify $$\\frac{x^{3}}{3}-x+c_{1}-y-\\frac{x^{3}}{3}+x-c_{1}+y:{\\quad}0$$", "input": "\\frac{x^{3}}{3}-x+c_{1}-y-\\frac{x^{3}}{3}+x-c_{1}+y", "result": "=e^{0}", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=-x+x+\\frac{x^{3}}{3}-\\frac{x^{3}}{3}+c_{1}-c_{1}-y+y" }, { "type": "step", "primary": "Add similar elements: $$\\frac{x^{3}}{3}-\\frac{x^{3}}{3}=0$$", "result": "=-x+x+c_{1}-c_{1}-y+y" }, { "type": "step", "primary": "Add similar elements: $$c_{1}-c_{1}=0$$", "result": "=-x+x-y+y" }, { "type": "step", "primary": "Add similar elements: $$-x+x=0$$", "result": "=-y+y" }, { "type": "step", "primary": "Add similar elements: $$-y+y=0$$", "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$a^{0}=1,\\:a\\ne\\:0$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WfSAAJsrV+Iw+rhsmhuMIKktL6TItPhEW0SBEr/UcGIpEvTz9ZL1nZ9Gtz6/xwjzXGDsOcvi27ad27I/2+uagC061ljBSPJeENOw2efoSWs8auWUd4WNoosLPjGkhvjRRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6r838JyacuBK8v/PvtmFzv/uI8fHO0kukCxXoJDmewy05BFoewEJfC9HKO/5L9BED6wbietrpeZmfGvxP+ADadL" } }, { "type": "step", "result": "ye^{-\\frac{x^{3}}{3}+x-c_{1}+y}=1" }, { "type": "step", "primary": "Divide both sides by $$e^{\\frac{-x^{3}+3x-3c_{1}}{3}}$$", "result": "\\frac{ye^{-\\frac{x^{3}}{3}+x-c_{1}+y}}{e^{\\frac{-x^{3}+3x-3c_{1}}{3}}}=\\frac{1}{e^{\\frac{-x^{3}+3x-3c_{1}}{3}}}" }, { "type": "interim", "title": "Simplify $$\\frac{ye^{-\\frac{x^{3}}{3}+x-c_{1}+y}}{e^{\\frac{-x^{3}+3x-3c_{1}}{3}}}:{\\quad}e^{y}y$$", "input": "\\frac{ye^{-\\frac{x^{3}}{3}+x-c_{1}+y}}{e^{\\frac{-x^{3}+3x-3c_{1}}{3}}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}\\:=\\:x^{a-b}$$", "secondary": [ "$$\\frac{e^{x-\\frac{x^{3}}{3}+y-c_{1}}}{e^{\\frac{-x^{3}+3x-3c_{1}}{3}}}=e^{\\left(x-\\frac{x^{3}}{3}+y-c_{1}\\right)-\\frac{-x^{3}+3x-3c_{1}}{3}}=e^{y}$$" ], "result": "=e^{y}y", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74EEBhPBJzpbKqZeFI3iemUO1jm469UfthrNrApjIhmKWdPDGglI4CKmYCvnxyBPDbeVxPeQ+jPt9jmuP2tK2QS/tNKIsWw9AVhwj3yl7J1Ddd47a0hQ8flDbGsI5To1d/gQNsw7O1R5nWGWBgyHXLaN6Hv6MoTMtvtU0IQwXdn9szOhN37mcRdV5CgGGkVwgD792h+X+WU95p+evzXAZDukexQm4asAoBYIY+Td3Voi6VUbFm4NaZn0kAJy18ULbLvpO9k03VfVPGUd16ZP8XxwJjQE2+jdHPi4NS7W+Vv0=" } }, { "type": "step", "result": "e^{y}y=\\frac{1}{e^{\\frac{-x^{3}+3x-3c_{1}}{3}}}" } ], "meta": { "interimType": "Equation Lambert Rewrite Title 1Eq" } }, { "type": "interim", "title": "Solve $$e^{y}y=\\frac{1}{e^{\\frac{-x^{3}+3x-3c_{1}}{3}}}:{\\quad}y=\\W\\left(\\frac{1}{e^{\\frac{-x^{3}+3x-3c_{1}}{3}}}\\right)$$", "input": "e^{y}y=\\frac{1}{e^{\\frac{-x^{3}+3x-3c_{1}}{3}}}", "steps": [ { "type": "step", "primary": "General solution for $$xe^x=a\\:$$is multivalued Lambert W function: $$x=\\W\\left(a\\right)$$", "result": "y=\\W\\left(\\frac{1}{e^{\\frac{-x^{3}+3x-3c_{1}}{3}}}\\right)" } ], "meta": { "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "y=\\W\\left(\\frac{1}{e^{\\frac{-x^{3}+3x-3c_{1}}{3}}}\\right)" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Isolate Title 1Eq" } }, { "type": "step", "primary": "Simplify", "result": "y=\\W\\left(\\frac{1}{e^{\\frac{-x^{3}+3x-c_{1}}{3}}}\\right)" } ], "meta": { "interimType": "ODE Solve Separable 0Eq" } }, { "type": "step", "result": "y=\\W\\left(\\frac{1}{e^{\\frac{-x^{3}+3x-c_{1}}{3}}}\\right)" } ], "meta": { "solvingClass": "ODE" } }, "meta": { "showVerify": true } }