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Question Number 169053 by Mastermind last updated on 23/Apr/22

$$SolvetheODE$$$$y^{\prime }+2xy={xe}^{-x^2},withy\left(0\right)=1$$$$$$$$Mastermind$$

Answered by haladu last updated on 23/Apr/22

$${\mathbf{y}}^{\prime }+\mathbf{p}\left(\mathbf{x}\right)\mathbf{y}=\mathbf{Q}\left(\mathbf{x}\right)$$$$\mathbf{p}\left(\mathbf{x}\right)=2\mathbf{x}$$$$$$$$\mathbf{I}.\mathbf{f}={\mathbf{e}}^{\int 2\mathbf{x}\mathbf{dx}}={\mathbf{e}}^{{\mathbf{x}}^2}$$$$$$$$\mathbf{y}{\mathbf{e}}^{{\mathbf{x}}^2}=\int {\mathbf{xe}}^{-{\mathbf{x}}^2}\times {\mathbf{e}}^{{\mathbf{x}}^2}\mathbf{dx}$$$$$$$${\mathbf{ye}}^{{\mathbf{x}}^2}=\int \mathbf{x}\mathbf{dx}=\frac{{\mathbf{x}}^2}{2}+\mathbf{C}$$$$$$$$\mathbf{y}={\mathbf{e}}^{-{\mathbf{x}}^2}(\frac{{\mathbf{x}}^2}{2}+\mathbf{C})$$$$$$$$\mathbf{x}=0\mathbf{y}=1$$$$$$$$1=\mathbf{C}$$$$$$$$\Rightarrow \mathbf{y}={\mathbf{e}}^{-{\mathbf{x}}^2}(\frac{{\mathbf{x}}^2}{2}+1)$$$$$$$$\mathbf{Haladu}\mathbf{B}\mathbf{Bello}$$

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