Answer to Question #318127 in Differential Equations for n hasan

Question #318127

Find the linear and Bernoulli’s differential equations from the following differential equations and solve it.

i)          (1 − 𝗑2) 𝑑𝑦 − 𝗑𝑦 = 1.

ii)         𝑑𝑦/dx = 𝗑𝑦2 − 𝗑𝑦.


 


1
Expert's answer
2022-03-28T12:10:37-0400

"i:\\\\\\left( 1-x^2 \\right) y'-xy=1\\\\y'-\\frac{x}{1-x^2}y=\\frac{1}{1-x^2}-linear\\,\\,equation\\\\P\\left( x \\right) =-\\frac{x}{1-x^2},Q\\left( x \\right) =\\frac{1}{1-x^2}\\\\y=\\frac{1}{\\exp \\left( \\int{Pdx} \\right)}\\left( \\int{Qe^{\\int{Pdx}}dx}+C \\right) =\\\\=\\frac{1}{\\exp \\left( \\int{\\frac{x}{x^2-1}dx} \\right)}\\left( \\int{\\frac{1}{1-x^2}e^{\\int{\\frac{x}{x^2-1}dx}}dx}+C \\right) =\\\\=\\frac{1}{\\exp \\left( \\frac{1}{2}\\ln \\left( 1-x^2 \\right) \\right)}\\left( \\int{\\frac{1}{1-x^2}e^{\\frac{1}{2}\\ln \\left( 1-x^2 \\right)}dx}+C \\right) =\\\\=\\frac{1}{\\sqrt{1-x^2}}\\left( \\int{\\frac{1}{\\sqrt{1-x^2}}dx}+C \\right) =\\frac{1}{\\sqrt{1-x^2}}\\left( \\mathrm{a}\\sin x+C \\right) \\\\ii:\\\\y'+xy=xy^2-Bernoulli\\,\\,equation\\\\y=0 -\\,\\,trivial\\\\P\\left( x \\right) =x,Q\\left( x \\right) =x,n=2\\\\u=y^{-1}\\\\y=\\frac{1}{u}\\\\y'=-\\frac{1}{u^2}u'\\\\-\\frac{1}{u^2}u'+\\frac{x}{u}=\\frac{x}{u^2}\\\\u'-xu=-x\\\\u'=x\\left( u-1 \\right) \\\\\\frac{d\\left( u-1 \\right)}{u-1}=xdx\\\\\\ln \\left| u-1 \\right|=\\frac{x^2}{2}\\\\u=Ce^{\\frac{x^2}{2}}+1\\\\y=\\frac{1}{Ce^{\\frac{x^2}{2}}+1}\\\\"


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