Question and Answers Forum

All Questions      Topic List

Differential Equation Questions

Previous in All Question      Next in All Question      

Previous in Differential Equation      Next in Differential Equation      

Question Number 169050 by Mastermind last updated on 23/Apr/22

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

Answered by Mathspace last updated on 23/Apr/22

$$h\to y^{{}^{\prime }}=-xy\Rightarrow \frac{y^{{}^{\prime }}}{y}=-x\Rightarrow$$$$ln\mid y\mid =-\frac{x^2}{2}+\lambda \Rightarrow y=k{e}^{-\frac{x^2}{2}}$$$$\left(mvc\right)\to y^{{}^{\prime }}=k^{{}^{\prime }}{e}^{-\frac{x^2}{2}}-kx{e}^{-\frac{x^2}{2}}$$$$\left(e\right)\Rightarrow k^{{}^{\prime }}{e}^{-\frac{x^2}{2}}-kx{e}^{-\frac{x^2}{2}}+xk{e}^{-\frac{x^2}{2}}=x^2$$$$\Rightarrow k^{{}^{\prime }}=x^2{e}^{\frac{x^2}{2}}\Rightarrow k={\int }_0^x{t}^2{e}^{\frac{t^2}{2}}dt+\lambda$$$$y\left(x\right)=(\lambda +{\int }_0^x{t}^2{e}^{\frac{t^2}{2}}dt)e^{-\frac{x^2}{2}}$$$$y\left(0\right)=2\Rightarrow \lambda =2\Rightarrow$$$$y\left(x\right)=2e^{-\frac{x^2}{2}}+e^{-\frac{x^2}{2}}{\int }_0^x{t}^2{e}^{\frac{t^2}{2}}dt$$

Commented by Mastermind last updated on 24/Apr/22

$$Ididnotunderstandthissir,$$$$couldyousolveitinanotherway$$$$$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com