-3y^'+9x^2=0

{ "query": { "display": "$$-3y^{^{\\prime}}+9x^{2}=0$$", "symbolab_question": "ODE#-3y^{\\prime }+9x^{2}=0" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "ODE", "subTopic": "LinearSimple", "default": "y=x^{3}+c_{1}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$-3y^{\\prime}\\left(x\\right)+9x^{2}=0:{\\quad}y=x^{3}+c_{1}$$", "input": "-3y^{\\prime}\\left(x\\right)+9x^{2}=0", "steps": [ { "type": "step" }, { "type": "interim", "title": "Isolate $$y^{\\prime}\\left(x\\right):{\\quad}y^{\\prime}\\left(x\\right)=3x^{2}$$", "input": "-3y^{\\prime}\\left(x\\right)+9x^{2}=0", "steps": [ { "type": "interim", "title": "Move $$9x^{2}\\:$$to the right side", "input": "-3y^{\\prime}\\left(x\\right)+9x^{2}=0", "result": "-3y^{\\prime}\\left(x\\right)=-9x^{2}", "steps": [ { "type": "step", "primary": "Subtract $$9x^{2}$$ from both sides", "result": "-3y^{^{\\prime}}\\left(x\\right)+9x^{2}-9x^{2}=0-9x^{2}" }, { "type": "step", "primary": "Simplify", "result": "-3y^{^{\\prime}}\\left(x\\right)=-9x^{2}" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-3$$", "input": "-3y^{\\prime}\\left(x\\right)=-9x^{2}", "result": "y^{\\prime}\\left(x\\right)=3x^{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$-3$$", "result": "\\frac{-3y^{^{\\prime}}\\left(x\\right)}{-3}=\\frac{-9x^{2}}{-3}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{-3y^{\\prime}\\left(x\\right)}{-3}=\\frac{-9x^{2}}{-3}", "result": "y^{\\prime}\\left(x\\right)=3x^{2}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{-3y^{\\prime}\\left(x\\right)}{-3}:{\\quad}y^{\\prime}\\left(x\\right)$$", "input": "\\frac{-3y^{\\prime}\\left(x\\right)}{-3}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{3y^{^{\\prime}}\\left(x\\right)}{3}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=y^{^{\\prime}}\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tSYX3wueu5hNA1A3JWz20QyQVoEnFfKeFM381ke9d4YgJ/ZZA32ZInFBpDtxBfiKc7NbROBSvSSByViggDVbGUq6gwX7tPb330QtIT+Or4CLGmNnLPWGf9PH3lpmjoJIFWPpL4Qe0FnKBJSzTcgMHbRW4/pfetOEUcv3CIro4N5XZeNU0R6SHwm0bdlcOVNB" } }, { "type": "interim", "title": "Simplify $$\\frac{-9x^{2}}{-3}:{\\quad}3x^{2}$$", "input": "\\frac{-9x^{2}}{-3}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{9x^{2}}{3}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{9}{3}=3$$", "result": "=3x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7X6Isc8SI2163OWL/ZETdv8YsRis5nMbjxpUVXVw8BRJwkKGJWEPFPk38sdJMsyPIQ6swNKnz+hXtZJQQrjYPz3ql8XXPq6bNQlMm+36iNhljgtURsNZ8mF2q2lQDr86HNy7T+bNZN9gkGieeklZaviS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "y^{^{\\prime}}\\left(x\\right)=3x^{2}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Isolate Title 1Eq" } }, { "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": "y=\\int\\:3x^{2}dx" }, { "type": "interim", "title": "$$\\int\\:3x^{2}dx=x^{3}+c_{1}$$", "input": "\\int\\:3x^{2}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": "=3\\cdot\\:\\int\\:x^{2}dx" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:x^{2}dx", "result": "=3\\cdot\\:\\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" } }, { "type": "interim", "title": "Simplify $$3\\cdot\\:\\frac{x^{3}}{3}:{\\quad}x^{3}$$", "input": "3\\cdot\\:\\frac{x^{3}}{3}", "result": "=x^{3}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{x^{3}\\cdot\\:3}{3}" }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=x^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CzXS+MvaOUdKL6IkXGMFAw9rc4FPHtW0FMZdPNL2pxnNGoPE9TME3q+OPmgkv2RQ3DEfvSyy0yowd6nPoCQ9C+9sGZu5A1MXROmEpnxG69rWcTN4w49IeAiiPupUjF67mCdVHcS+WNyfofhuXmao2p0r1PCayV8M6PZcfTYBjkw=" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=x^{3}+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": "y=x^{3}+c_{1}" } ], "meta": { "solvingClass": "ODE" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "#>#ODE#>#y=x^{3}+c_{1}" } } }, "meta": { "showVerify": true } }