(dy)/(dx)=e^{x+y+3}

{ "query": { "display": "$$\\frac{dy}{dx}=e^{x+y+3}$$", "symbolab_question": "ODE#\\frac{dy}{dx}=e^{x+y+3}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "ODE", "subTopic": "FirstSeparable", "default": "y=-\\ln(-e^{3+x}-c_{1})", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{dy}{dx}=e^{x+y+3}:{\\quad}y=-\\ln\\left(-e^{3+x}-c_{1}\\right)$$", "input": "\\frac{dy}{dx}=e^{x+y+3}", "steps": [ { "type": "interim", "title": "Solve separable ODE:$${\\quad}y=-\\ln\\left(-e^{3+x}-c_{1}\\right)$$", "input": "\\frac{dy}{dx}=e^{x+y+3}", "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": "Substitute $$\\frac{dy}{dx}$$ with $$y^{\\prime}\\left(x\\right)$$", "result": "y^{^{\\prime}}\\left(x\\right)=e^{x+y+3}" }, { "type": "interim", "title": "Rewrite in the form of a first order separable ODE", "input": "y^{\\prime}\\left(x\\right)=e^{x+y+3}", "result": "\\frac{1}{e^{y}}y^{\\prime}\\left(x\\right)=e^{x+3}", "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)=e^{x+y+3}" }, { "type": "step", "primary": "Divide both sides by $$e^{y}$$", "result": "\\frac{y^{^{\\prime}}\\left(x\\right)}{e^{y}}=\\frac{e^{x+y+3}}{e^{y}}" }, { "type": "step", "primary": "Simplify", "result": "\\frac{y^{^{\\prime}}\\left(x\\right)}{e^{y}}=e^{x+3}" }, { "type": "step", "primary": "Rewrite in standard form", "secondary": [ "$$N\\left(y\\right)=\\frac{1}{e^{y}},\\:{\\quad}M\\left(x\\right)=e^{x+3}$$" ], "result": "\\frac{1}{e^{y}}y^{^{\\prime}}\\left(x\\right)=e^{x+3}" } ], "meta": { "interimType": "Canon First Order Separable ODE 2Eq" } }, { "type": "interim", "title": "Solve $$\\frac{1}{e^{y}}y^{\\prime}\\left(x\\right)=e^{x+3}:{\\quad}-\\frac{1}{e^{y}}=e^{3+x}+c_{1}$$", "input": "\\frac{1}{e^{y}}y^{\\prime}\\left(x\\right)=e^{x+3}", "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}{e^{y}}dy=\\int\\:e^{x+3}dx" }, { "type": "step", "primary": "Integrate each side of the equation" }, { "type": "interim", "title": "$$\\int\\:e^{x+3}dx=e^{3+x}+c_{1}$$", "input": "\\int\\:e^{x+3}dx", "steps": [ { "type": "step", "primary": "$$e^{x+3}=e^{3}e^{x}$$", "result": "=\\int\\:e^{3}e^{x}dx" }, { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=e^{3}\\cdot\\:\\int\\:e^{x}dx" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:e^{x}dx=e^{x}$$", "result": "=e^{3}e^{x}" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$e^{3}e^{x}=\\:e^{3+x}$$" ], "result": "=e^{3+x}", "meta": { "solvingClass": "Solver", "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=e^{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": "$$\\int\\:\\frac{1}{e^{y}}dy=-\\frac{1}{e^{y}}+c_{2}$$", "input": "\\int\\:\\frac{1}{e^{y}}dy", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{1}{e^{y}}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: $$u=e^{y}$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dy}=e^{y}$$", "input": "\\left(e^{y}\\right)^{^{\\prime}}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\left(e^{y}\\right)^{\\prime}=e^{y}$$", "result": "=e^{y}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7L+rjPEvTTKt9OQYXca46gG1efk4K/SCcwkEOFfnqcSqXIQHgliMhSOSNsNni19Inh3lRwhsm/xG+3FxBuJ667aN6Hv6MoTMtvtU0IQwXdn9cF1dPzrDZNZLtq7Ed4KEurrwiBA+mZRVs2zU1cnrEdA==" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=e^{y}dy$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dy=\\frac{1}{e^{y}}du$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{u}\\cdot\\:\\frac{1}{e^{y}}du" }, { "type": "step", "primary": "$$u=e^{y}$$", "result": "=\\int\\:\\frac{1}{u}\\cdot\\:\\frac{1}{u}du" }, { "type": "interim", "title": "Simplify $$\\frac{1}{u}\\cdot\\:\\frac{1}{u}:{\\quad}\\frac{1}{u^{2}}$$", "input": "\\frac{1}{u}\\cdot\\:\\frac{1}{u}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{1\\cdot\\:1}{uu}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=\\frac{1}{uu}" }, { "type": "interim", "title": "$$uu=u^{2}$$", "input": "uu", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$uu=\\:u^{1+1}$$" ], "result": "=u^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/E93FIYHpq26Gj2mwfeoqMzBWJotReR4P4m6RE6FZ2Oes25OoAq8kAwRqD36EFJe4ylfb0DUJOE0oSeuKQ0IOSS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=\\frac{1}{u^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{u^{2}}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7+RgdgxjYl30Sar+nMkR9V4qzJGeyM6S0DCrZisZSya8/6umhM84nJGixAkjaEU36WyJTZPfTsR1UcTUz/dkJr37zLo4mNmPa+Gfhzdrj9zJocZM7vUW/A7/O1+TQrSRTqN6Hv6MoTMtvtU0IQwXdn84k+SM9uK5ZgVIsXdnKy6Jugs8OYsrd0uK6mBpcEuDFg==" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{u^{2}}du" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:\\frac{1}{u^{2}}du", "result": "=-\\frac{1}{u}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a^b}=a^{-b}$$", "secondary": [ "$$\\frac{1}{u^{2}}=u^{-2}$$" ], "result": "=\\int\\:u^{-2}du", "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{u^{-2+1}}{-2+1}" }, { "type": "interim", "title": "Simplify $$\\frac{u^{-2+1}}{-2+1}:{\\quad}-\\frac{1}{u}$$", "input": "\\frac{u^{-2+1}}{-2+1}", "steps": [ { "type": "step", "primary": "Add/Subtract the numbers: $$-2+1=-1$$", "result": "=\\frac{u^{-1}}{-1}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{u^{-1}}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=-u^{-1}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-1}=\\frac{1}{a}$$", "result": "=-\\frac{1}{u}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=-\\frac{1}{u}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7/RSr02Agv0MR/qV7Nm+eMMy4+rY5ULRUEksemusM4Yyrrf9ZAnPXwtHEGeHjeiUc8XwLUgD2yVoFe9iCfntTx4OQzbEnsuafNY3nX9QxDlJ1HXTSqqQEjS1gpf6I+JyHQS4M5VpC8qh+oehjmM1qmweKkh+28FiXwy+Vsz8xLQiialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "step", "primary": "Substitute back $$u=e^{y}$$", "result": "=-\\frac{1}{e^{y}}" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-\\frac{1}{e^{y}}+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}{e^{y}}+c_{2}=e^{3+x}+c_{1}" }, { "type": "step", "primary": "Combine the constants", "result": "-\\frac{1}{e^{y}}=e^{3+x}+c_{1}" } ], "meta": { "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Isolate $$y:{\\quad}y=-\\ln\\left(-e^{3+x}-c_{1}\\right)$$", "input": "-\\frac{1}{e^{y}}=e^{3+x}+c_{1}", "steps": [ { "type": "interim", "title": "Apply exponent rules", "input": "-\\frac{1}{e^{y}}=e^{3+x}+c_{1}", "result": "-y=\\ln\\left(-e^{3+x}-c_{1}\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a^b}=a^{-b}$$", "secondary": [ "$$\\frac{1}{e^{y}}=e^{-y}$$" ], "result": "-e^{-y}=e^{3+x}+c_{1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "Divide both sides by $$-1$$", "input": "-e^{-y}=e^{3+x}+c_{1}", "result": "e^{-y}=-e^{3+x}-c_{1}", "steps": [ { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "\\frac{-e^{-y}}{-1}=\\frac{e^{3+x}}{-1}+\\frac{c_{1}}{-1}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{-e^{-y}}{-1}=\\frac{e^{3+x}}{-1}+\\frac{c_{1}}{-1}", "result": "e^{-y}=-e^{3+x}-c_{1}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{-e^{-y}}{-1}:{\\quad}e^{-y}$$", "input": "\\frac{-e^{-y}}{-1}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{e^{-y}}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=e^{-y}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7pccl4v35CSIRZyFm9e4CQr6cnqShpGbg4GQApXSZWPtwkKGJWEPFPk38sdJMsyPIflzQf456rkvV6kPSp58/Onql8XXPq6bNQlMm+36iNhljgtURsNZ8mF2q2lQDr86HSl3BG13qcfaIj6JZROKIeCS3daIZHtloJpe/PvtsyNI=" } }, { "type": "interim", "title": "Simplify $$\\frac{e^{3+x}}{-1}+\\frac{c_{1}}{-1}:{\\quad}-e^{3+x}-c_{1}$$", "input": "\\frac{e^{3+x}}{-1}+\\frac{c_{1}}{-1}", "steps": [ { "type": "interim", "title": "$$\\frac{e^{3+x}}{-1}=-e^{3+x}$$", "input": "\\frac{e^{3+x}}{-1}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{e^{3+x}}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=-e^{3+x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s772EgzmCZ+NwcB/UlDmKB176cnqShpGbg4GQApXSZWPtwkKGJWEPFPk38sdJMsyPIe+DjCjROwUUD6Z7s1yK6qs1bIZxfodm3UsZcfZAZr4uft/AZlh4VHsL88X9Is3CnTr0dgwyLhjT78psIFYgsdQ==" } }, { "type": "step", "result": "=-e^{3+x}+\\frac{c_{1}}{-1}" }, { "type": "interim", "title": "$$\\frac{c_{1}}{-1}=-c_{1}$$", "input": "\\frac{c_{1}}{-1}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{c_{1}}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=-c_{1}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OFapZl6h6k27790x8zBueKiJTJQkRRngZl07rbqjeC6jkVi15I8rBefLi4Iyt2wrSOkhs11OGwlxoLjr/XtsUvIUSkMFJCI6Q2t3EMyW7beEuxdw3HgFt45DBKCVsPjaJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=-e^{3+x}-c_{1}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s772EgzmCZ+NwcB/UlDmKB1wMJ8wJzszdK9zf8DjikF2BTZY26BaOrKvHNmwlxGSfgCUCWbkwGOY7PqKo3U/JLJRjtiiR5tm+6rzzY8CX3Y/QWCGEwNT8ujLuXh+hNVyJYHimBRYRqHSWeJkuUPhfTC5hKkN1ZySefULyQnmbmuVBtMXRuigsiew2T9nad3rUivnCbpjiD3mWQRnrOLGxDsg==" } }, { "type": "step", "result": "e^{-y}=-e^{3+x}-c_{1}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "If $$f\\left(x\\right)=g\\left(x\\right)$$, then $$\\ln\\left(f\\left(x\\right)\\right)=\\ln\\left(g\\left(x\\right)\\right)$$", "result": "\\ln\\left(e^{-y}\\right)=\\ln\\left(-e^{3+x}-c_{1}\\right)" }, { "type": "step", "primary": "Apply log rule: $$\\ln\\left(e^a\\right)=a$$", "secondary": [ "$$\\ln\\left(e^{-y}\\right)=-y$$" ], "result": "-y=\\ln\\left(-e^{3+x}-c_{1}\\right)", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Apply Exp Rules Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74TqdRGDyv3oqElK9EevGS4d9JdzB0roV8xH/k/xXQ/olyfrojFMyj6/jYQtPZ+6/q97FaqgAnVMIJDZuyKY89qOMbxmnb9QvXvL8exTiek78PiInH+V+YrFRieQQ0uTcikVqvxSB4FQSZm/NxkIHjA/BYw+yh08MsdrktBTYf2Dmw1q+hjFEjKzJRgYLTRGSdb8L0DQCV+SJS0KAGgZzMYZ+y+YEwVMxeIPlP7C0eko=" } }, { "type": "interim", "title": "Solve $$-y=\\ln\\left(-e^{3+x}-c_{1}\\right):{\\quad}y=-\\ln\\left(-e^{3+x}-c_{1}\\right)$$", "input": "-y=\\ln\\left(-e^{3+x}-c_{1}\\right)", "steps": [ { "type": "interim", "title": "Divide both sides by $$-1$$", "input": "-y=\\ln\\left(-e^{3+x}-c_{1}\\right)", "result": "y=-\\ln\\left(-e^{3+x}-c_{1}\\right)", "steps": [ { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "\\frac{-y}{-1}=\\frac{\\ln\\left(-e^{3+x}-c_{1}\\right)}{-1}" }, { "type": "step", "primary": "Simplify", "result": "y=-\\ln\\left(-e^{3+x}-c_{1}\\right)" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "y=-\\ln\\left(-e^{3+x}-c_{1}\\right)" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Isolate Title 1Eq" } }, { "type": "step", "result": "y=-\\ln\\left(-e^{3+x}-c_{1}\\right)" } ], "meta": { "interimType": "ODE Solve Separable 0Eq" } }, { "type": "step", "result": "y=-\\ln\\left(-e^{3+x}-c_{1}\\right)" } ], "meta": { "solvingClass": "ODE" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "#>#ODE#>#y=-\\ln(-e^{3+x}-c_{1})" } } }, "meta": { "showVerify": true } }