What is the general solution of the differential equation? : $\frac{dy}{dx}+3y=3{x}^{2}{e}^{-3x}$

Liam Keller
2022-09-13
Answered

What is the general solution of the differential equation? : $\frac{dy}{dx}+3y=3{x}^{2}{e}^{-3x}$

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asked 2022-06-16

Given differential equations

$\ddot{x}=G{m}_{1}\frac{y-x}{|y-x{|}^{3}}\phantom{\rule{2cm}{0ex}}\ddot{y}=G{m}_{2}\frac{x-y}{|y-x{|}^{3}}$

with constant G,m1,m2 I want to solve them with the Euler method. I know I have to reduce the equations to a system of first order. So I did for $\ddot{x}$

$\frac{d}{dt}\left(\begin{array}{c}{v}_{0}\\ {v}_{1}\end{array}\right)=\left(\begin{array}{c}{v}_{1}\\ G{m}_{1}\frac{y-{v}_{0}}{|{v}_{0}-y{|}^{3}}\end{array}\right)$

and the same for $\ddot{y}$. But now my problem is that the right side depends on the other equation and the other way round. How can you get a system of first order so you can apply Euler to it?

$\ddot{x}=G{m}_{1}\frac{y-x}{|y-x{|}^{3}}\phantom{\rule{2cm}{0ex}}\ddot{y}=G{m}_{2}\frac{x-y}{|y-x{|}^{3}}$

with constant G,m1,m2 I want to solve them with the Euler method. I know I have to reduce the equations to a system of first order. So I did for $\ddot{x}$

$\frac{d}{dt}\left(\begin{array}{c}{v}_{0}\\ {v}_{1}\end{array}\right)=\left(\begin{array}{c}{v}_{1}\\ G{m}_{1}\frac{y-{v}_{0}}{|{v}_{0}-y{|}^{3}}\end{array}\right)$

and the same for $\ddot{y}$. But now my problem is that the right side depends on the other equation and the other way round. How can you get a system of first order so you can apply Euler to it?

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