y^'=0.5(3-y)

{ "query": { "display": "$$y^{^{\\prime}}=0.5\\left(3-y\\right)$$", "symbolab_question": "ODE#y^{\\prime }=0.5(3-y)" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "ODE", "subTopic": "FirstSeparable", "default": "y=-e^{-\\frac{t}{2}-c_{1}}+3", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$y^{\\prime}\\left(t\\right)=0.5\\left(3-y\\right):{\\quad}y=-e^{-\\frac{t}{2}-c_{1}}+3$$", "input": "y^{\\prime}\\left(t\\right)=0.5\\left(3-y\\right)", "steps": [ { "type": "interim", "title": "Solve separable ODE:$${\\quad}y=-e^{-\\frac{t}{2}-c_{1}}+3$$", "input": "y^{\\prime}\\left(t\\right)=0.5\\left(3-y\\right)", "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(t\\right)=0.5\\left(3-y\\right)", "result": "\\frac{1}{3-y}y^{\\prime}\\left(t\\right)=0.5", "steps": [ { "type": "step", "primary": "Standard form of a first order separable ODE:", "secondary": [ "$$N\\left(y\\right){\\cdot}y^{\\prime}\\left(t\\right)=M\\left(t\\right)$$" ] }, { "type": "step", "result": "y^{^{\\prime}}\\left(t\\right)=0.5\\left(3-y\\right)" }, { "type": "step", "primary": "Divide both sides by $$3-y$$", "result": "\\frac{y^{^{\\prime}}\\left(t\\right)}{3-y}=\\frac{0.5\\left(3-y\\right)}{3-y}" }, { "type": "step", "primary": "Simplify", "result": "\\frac{y^{^{\\prime}}\\left(t\\right)}{3-y}=0.5" }, { "type": "step", "primary": "Rewrite in standard form", "secondary": [ "$$N\\left(y\\right)=\\frac{1}{3-y},\\:{\\quad}M\\left(t\\right)=0.5$$" ], "result": "\\frac{1}{3-y}y^{^{\\prime}}\\left(t\\right)=0.5" } ], "meta": { "interimType": "Canon First Order Separable ODE 2Eq" } }, { "type": "interim", "title": "Solve $$\\frac{1}{3-y}y^{\\prime}\\left(t\\right)=0.5:{\\quad}-\\ln\\left(3-y\\right)=0.5t+c_{1}$$", "input": "\\frac{1}{3-y}y^{\\prime}\\left(t\\right)=0.5", "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}{3-y}dy=\\int\\:0.5dt" }, { "type": "step", "primary": "Integrate each side of the equation" }, { "type": "interim", "title": "$$\\int\\:0.5dt=0.5t+c_{1}$$", "input": "\\int\\:0.5dt", "steps": [ { "type": "step", "primary": "Integral of a constant: $$\\int{a}dx=ax$$", "result": "=0.5t" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=0.5t+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": "$$\\int\\:\\frac{1}{3-y}dy=-\\ln\\left(3-y\\right)+c_{2}$$", "input": "\\int\\:\\frac{1}{3-y}dy", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{1}{3-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: $$u=3-y$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dy}=-1$$", "input": "\\left(3-y\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=3^{^{\\prime}}-y^{^{\\prime}}\\left(t\\right)" }, { "type": "interim", "title": "$$3^{\\prime}=0$$", "input": "3^{\\prime}", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+nIdIwylUMYOMO9RBaTZQqboRT4ICWDw+vXUOyCoK0ujkVi15I8rBefLi4Iyt2wr8D4yaPBYvrqNvcxJbQLVhFZj9jt8zUuYsToWb4jt0JM=" } }, { "type": "interim", "title": "$$y^{\\prime}=1$$", "input": "y^{\\prime}\\left(t\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$y^{\\prime}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AdXIoCyPf2+1nCWNuVlZKOPYLix+cHWJS0/uAuSicfOQuIxj9K+Upo9l4tAcwN/gE9/03SOiEv+BIHutWLr6nUYzcBSAZ7Unbh0Vq2yWIiO/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "result": "=0-1" }, { "type": "step", "primary": "Simplify", "result": "=-1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=-1dy$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dy=\\left(-1\\right)du$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{u}\\left(-1\\right)du" }, { "type": "step", "result": "=\\int\\:-\\frac{1}{u}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s74MGWJpmcz1rfOXR+/Hh6kgcjlLRK1jUV206qo4+vRN7yRnKSRJBHsnEXq1wS/zww7h8KUX9hN21wJE16H/NfZwoJk9nAKBDt2je44rU15E3IRavGseoPlXpKuGEbhS3N9bA+zX4bD3u3gx65o2NJhOP8jPOjUAODCISjG3EPUUsKIcdyTU2qbTbjVtrM8zqhQ==" } }, { "type": "step", "result": "=\\int\\:-\\frac{1}{u}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": "=-\\int\\:\\frac{1}{u}du" }, { "type": "step", "primary": "Use the common integral:$$\\quad\\:\\int\\:\\frac{1}{u}du=\\ln\\left(u\\right),\\:$$assuming a complex-valued logarithm", "result": "=-\\ln\\left(u\\right)" }, { "type": "step", "primary": "Substitute back $$u=3-y$$", "result": "=-\\ln\\left(3-y\\right)" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-\\ln\\left(3-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": "-\\ln\\left(3-y\\right)+c_{2}=0.5t+c_{1}" }, { "type": "step", "primary": "Combine the constants", "result": "-\\ln\\left(3-y\\right)=0.5t+c_{1}" } ], "meta": { "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Isolate $$y:{\\quad}y=-e^{-\\frac{t}{2}-c_{1}}+3$$", "input": "-\\ln\\left(3-y\\right)=0.5t+c_{1}", "steps": [ { "type": "interim", "title": "Multiply both sides by $$10$$", "input": "-\\ln\\left(3-y\\right)=0.5t+c_{1}", "result": "-\\ln\\left(3-y\\right)\\cdot\\:10=5t+10c_{1}", "steps": [ { "type": "step", "primary": "To eliminate decimal points, multiply by 10 for every digit after the decimal point", "secondary": [ "There is one digit to the right of the decimal point, therefore multiply by $$10$$" ], "result": "-\\ln\\left(3-y\\right)\\cdot\\:10=0.5t\\cdot\\:10+c_{1}\\cdot\\:10" }, { "type": "step", "primary": "Refine", "result": "-\\ln\\left(3-y\\right)\\cdot\\:10=5t+10c_{1}" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-10$$", "input": "-\\ln\\left(3-y\\right)\\cdot\\:10=5t+10c_{1}", "result": "\\ln\\left(3-y\\right)=-\\frac{t}{2}-c_{1}", "steps": [ { "type": "step", "primary": "Divide both sides by $$-10$$", "result": "\\frac{-\\ln\\left(3-y\\right)\\cdot\\:10}{-10}=\\frac{5t}{-10}+\\frac{10c_{1}}{-10}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{-\\ln\\left(3-y\\right)\\cdot\\:10}{-10}=\\frac{5t}{-10}+\\frac{10c_{1}}{-10}", "result": "\\ln\\left(3-y\\right)=-\\frac{t}{2}-c_{1}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{-\\ln\\left(3-y\\right)\\cdot\\:10}{-10}:{\\quad}\\ln\\left(3-y\\right)$$", "input": "\\frac{-\\ln\\left(3-y\\right)\\cdot\\:10}{-10}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{\\ln\\left(3-y\\right)\\cdot\\:10}{10}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{10}{10}=1$$", "result": "=\\ln\\left(3-y\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7k8reiFheuO5e5Tb8raNgVW8TmF05YJKIW6YNVFDP/n+g5HyicSWVQpnwVAL48qxJ/aL2Coo0GISQwm8bv5wDia8Kozg/yHGSox9EdceN6ZF6pfF1z6umzUJTJvt+ojYZY4LVEbDWfJhdqtpUA6/Oh7v6HEnMrW7wRGQeBTpDUdF3pWyfZPcAPZIXb77SOdbD" } }, { "type": "interim", "title": "Simplify $$\\frac{5t}{-10}+\\frac{10c_{1}}{-10}:{\\quad}-\\frac{t}{2}-c_{1}$$", "input": "\\frac{5t}{-10}+\\frac{10c_{1}}{-10}", "steps": [ { "type": "interim", "title": "$$\\frac{5t}{-10}=-\\frac{t}{2}$$", "input": "\\frac{5t}{-10}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{5t}{10}" }, { "type": "step", "primary": "Cancel the common factor: $$5$$", "result": "=-\\frac{t}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Uwqx3ibTep/8KZQoK16faHyRHuGw7+tM5METTDj6vVHDFYNeJ0RX0LOwY+dS7jXMvTBQfvyZjNWJlQl8cr6zPw4bfwiV6iMLJ5sC1nL7dObXuje6Ajf80mbD6FhDVp1VTksFedh1vOI72yYk7d5HfA==" } }, { "type": "step", "result": "=-\\frac{t}{2}+\\frac{10c_{1}}{-10}" }, { "type": "interim", "title": "$$\\frac{10c_{1}}{-10}=-c_{1}$$", "input": "\\frac{10c_{1}}{-10}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{10c_{1}}{10}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{10}{10}=1$$", "result": "=-c_{1}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s700ZYXhTGANG94s7BxXNF5dMcTiPU+yALgxiGfI2HHhyrju+5Z51e/ZZSD3gRHwjBbl2fHEzzOC+6bG6RDHLaQVNbbqpyK7JQEZdATEJR51g96MKgUdyzSEeogCDBBpBHqVA4qe1cnRxb7sWYxfAGag==" } }, { "type": "step", "result": "=-\\frac{t}{2}-c_{1}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Uwqx3ibTep/8KZQoK16faKH6A4OYyZkr6CungYXuv4PTHE4j1PsgC4MYhnyNhx4cq47vuWedXv2WUg94ER8IwVL/J79uAaXyV16l49SG/ldI6SGzXU4bCXGguOv9e2xSHjb2+5NLFZrsH9fcPWg/TcBZCZf4UBlDnt5DwfBsIkLEQm58NMD5Zm0LcMZkeJ5HAFrZpjH8LxZ99Wi0tA0PiA==" } }, { "type": "step", "result": "\\ln\\left(3-y\\right)=-\\frac{t}{2}-c_{1}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Apply log rules", "input": "\\ln\\left(3-y\\right)=-\\frac{t}{2}-c_{1}", "result": "3-y=e^{-\\frac{t}{2}-c_{1}}", "steps": [ { "type": "step", "primary": "Use the logarithmic definition: If $$\\log_a\\left(b\\right)=c\\:$$then $$b=a^c$$", "secondary": [ "$$\\ln\\left(3-y\\right)=-\\frac{t}{2}-c_{1}\\quad\\:\\Rightarrow\\:\\quad\\:3-y=e^{-\\frac{t}{2}-c_{1}}$$" ], "result": "3-y=e^{-\\frac{t}{2}-c_{1}}" } ], "meta": { "interimType": "Apply Log Rules Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/24TcuoVCfbg1c7qHcNzAnxOWR8PQvajPikrU8U35kPXcIbKuyGK1uwME4TodHAlwDpw3LFeH5YdRg6n4WygV7FQVdqvt7EX3qo1hxX6YE4Y6eGi+gMMryHZL1KA906tdFD56+MAFABfHeK1q5Mr3iP2FwLJ8fbD559jHjp+zXpSBv6izheLVUKQ/emokAUyzWgXvbCLLWf9Bt9k0yt9rg==" } }, { "type": "interim", "title": "Solve $$3-y=e^{-\\frac{t}{2}-c_{1}}:{\\quad}y=-e^{-\\frac{t}{2}-c_{1}}+3$$", "input": "3-y=e^{-\\frac{t}{2}-c_{1}}", "result": "y=-e^{-\\frac{t}{2}-c_{1}}+3", "steps": [ { "type": "interim", "title": "Move $$3\\:$$to the right side", "input": "3-y=e^{-\\frac{t}{2}-c_{1}}", "result": "-y=e^{-\\frac{t}{2}-c_{1}}-3", "steps": [ { "type": "step", "primary": "Subtract $$3$$ from both sides", "result": "3-y-3=e^{-\\frac{t}{2}-c_{1}}-3" }, { "type": "step", "primary": "Simplify", "result": "-y=e^{-\\frac{t}{2}-c_{1}}-3" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-1$$", "input": "-y=e^{-\\frac{t}{2}-c_{1}}-3", "result": "y=-e^{-\\frac{t}{2}-c_{1}}+3", "steps": [ { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "\\frac{-y}{-1}=\\frac{e^{-\\frac{t}{2}-c_{1}}}{-1}-\\frac{3}{-1}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{-y}{-1}=\\frac{e^{-\\frac{t}{2}-c_{1}}}{-1}-\\frac{3}{-1}", "result": "y=-e^{-\\frac{t}{2}-c_{1}}+3", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{-y}{-1}:{\\quad}y$$", "input": "\\frac{-y}{-1}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{y}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=y" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WfLNlsDM8wy8xk2WOanl7QCWKUbvV6WK3fDUgFtg3Q9YNuqLUawMjXCisTP0ZBbRo3oe/oyhMy2+1TQhDBd2f2zM6E3fuZxF1XkKAYaRXCAKmiwRFjGP7Y8/vx2spQ1zvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "interim", "title": "Simplify $$\\frac{e^{-\\frac{t}{2}-c_{1}}}{-1}-\\frac{3}{-1}:{\\quad}-e^{-\\frac{t}{2}-c_{1}}+3$$", "input": "\\frac{e^{-\\frac{t}{2}-c_{1}}}{-1}-\\frac{3}{-1}", "steps": [ { "type": "interim", "title": "$$\\frac{e^{-\\frac{t}{2}-c_{1}}}{-1}=-e^{-\\frac{t}{2}-c_{1}}$$", "input": "\\frac{e^{-\\frac{t}{2}-c_{1}}}{-1}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{e^{-\\frac{t}{2}-c_{1}}}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=-e^{-c_{1}-\\frac{t}{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QTQO75kZ51yAs3A0NZ159AxVf0otc6t5NB4PbYaop+O+nJ6koaRm4OBkAKV0mVj7cJChiVhDxT5N/LHSTLMjyI0LG9iDZbPC4qhX95XIqelB5CfuKT2cpcKiog4QgqbAzVshnF+h2bdSxlx9kBmvi4o11pd7YDFfIuqPWJIZaUU1bf3tp3BwbnmZLJazb5mb9+hfRmI4JIOrEGiVWyEVL53MkGqbnzQcO27TTIrxrZU=" } }, { "type": "step", "result": "=-e^{-c_{1}-\\frac{t}{2}}-\\frac{3}{-1}" }, { "type": "interim", "title": "$$\\frac{3}{-1}=-3$$", "input": "\\frac{3}{-1}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{3}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=-3" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OoyJX/1d2vs85gXbM5KP1S061ljBSPJeENOw2efoSWvW46RXHXnPJFddhpLU4qOFo3oe/oyhMy2+1TQhDBd2fwJtBms+ndTQ0d8s54OCeMEkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=-e^{-c_{1}-\\frac{t}{2}}-\\left(-3\\right)" }, { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=-e^{-\\frac{t}{2}-c_{1}}+3" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": 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