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Question Number 116136 by bemath last updated on 01/Oct/20

$${\mathrm{y}}^{\prime }=\frac{\mathrm{y}}{\mathrm{x}}+\frac{{2\mathrm{x}}^3\cos \left({\mathrm{x}}^2\right)}{\mathrm{y}}$$$$\mathrm{where}\mathrm{y}\left(\sqrt{\pi }\right)=0$$

Answered by mindispower last updated on 01/Oct/20

$$\frac{y}{x}=z$$$$y^{\prime }=z+{xz}^{\prime }$$$$\Rightarrow z+{xz}^{\prime }=z+\frac{2x^{3}cos\left(x^2\right)}{xz}$$$$\Rightarrow z\left({xz}^{\prime }\right)=2x^{2}cos\left(x^2\right)$$$$\Rightarrow {zz}^{\prime }=2xcos\left(x^2\right)$$$$\Rightarrow \int zdz=\int 2xcos\left(x^2\right)dx$$$$\iff \frac{z^2}{2}=sin\left(x^2\right)+c$$$$z=\underset{-}{+}2\sqrt{sin\left(x^2\right)+c}$$$$y=\underset{-}{+}z\sqrt{sin\left(x^2\right)+c}$$$$y\left(\sqrt{\pi }\right)=\underset{-}{+}2\sqrt{\pi }.\sqrt{sin\left(\pi \right)+c}=0\Rightarrow c=0$$$$y\left(x\right)=\underset{-}{+}2\sqrt{sin\left(x^2\right)}$$$$$$$$$$

Answered by TANMAY PANACEA last updated on 01/Oct/20

$$\frac{dy}{dx}-\frac{y}{x}=\frac{2x^3{cosx}^2}{y}$$$$\frac{xdy-ydx}{xdx\times x^2}=\frac{2{cosx}^2}{\left(\frac{y}{x}\right)}$$$$d\left(\frac{y}{x}\right).\frac{y}{x}={cosx}^2.{dx}^2$$$$intregating$$$$\int \left(\frac{y}{x}\right)d\left(\frac{y}{x}\right)=\int {cosx}^2.{dx}^2$$$$\frac{1}{2}{\left(\frac{y}{x}\right)}^2={sinx}^2+c$$

Answered by mathmax by abdo last updated on 01/Oct/20

$${\mathrm{y}}^{{}^{\prime }}=\frac{\mathrm{y}}{\mathrm{x}}+\frac{{2\mathrm{x}}^3\cos \left({\mathrm{x}}^2\right)}{\mathrm{y}}\mathrm{let}\frac{\mathrm{y}}{\mathrm{x}}=\mathrm{z}\Rightarrow \mathrm{y}=\mathrm{xz}\Rightarrow {\mathrm{y}}^{{}^{\prime }}=\mathrm{z}+{\mathrm{xz}}^{{}^{\prime }}$$$$\left(\mathrm{e}\right)\Rightarrow \mathrm{z}+{\mathrm{xz}}^{{}^{\prime }}=\mathrm{z}+\frac{{2\mathrm{x}}^2\cos \left({\mathrm{x}}^2\right)}{\mathrm{z}}\Rightarrow {\mathrm{xzz}}^{{}^{\prime }}={2\mathrm{x}}^2\cos \left({\mathrm{x}}^2\right)\Rightarrow$$$${\mathrm{zz}}^{{}^{\prime }}=2\mathrm{xcos}\left({\mathrm{x}}^2\right)\Rightarrow \int {\mathrm{zz}}^{{}^{\prime }}\mathrm{dx}=2\int \mathrm{xcos}\left({\mathrm{x}}^2\right)\mathrm{dx}=\sin \left({\mathrm{x}}^2\right)+\mathrm{c}\Rightarrow$$$$\frac{1}{2}{\mathrm{z}}^2=\sin \left({\mathrm{x}}^2\right)+\mathrm{c}\Rightarrow {\mathrm{z}}^2=2\sin \left({\mathrm{x}}^2\right)+2\mathrm{c}\Rightarrow \mathrm{z}=\stackrel{-}{+}\sqrt{2\sin \left({\mathrm{x}}^2\right)+\lambda }$$$$$$

Commented by mathmax by abdo last updated on 01/Oct/20

$$\Rightarrow \mathrm{y}=\stackrel{-}{+}\mathrm{x}\sqrt{2\sin \left({\mathrm{x}}^2\right)+\lambda }$$$$\mathrm{y}\left(\sqrt{\pi }\right)=0\Rightarrow \stackrel{-}{+}\sqrt{\pi }\sqrt{0+\lambda }=0\Rightarrow \lambda =0\Rightarrow \mathrm{y}=\stackrel{-}{+}\sqrt{2\sin \left({\mathrm{x}}^2\right)}$$

Commented by mathmax by abdo last updated on 02/Oct/20

$$\mathrm{y}=\stackrel{-}{+}\mathrm{x}\sqrt{2\sin \left({\mathrm{x}}^2\right)}$$

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