Question

find the general formula of
(dy/dx)+((2x+1)/x)*y=e^(-2x). y(1)=1/(e^2)

(i used ' ^ ' as power sign.)

Answer 1

Go to the math chat box and ill try to help you if you try to help me please please please

Answer 2

\[\frac{\mathrm{d}y}{\mathrm{d}x}+\frac{2x+1}{x}y=e^{-2x}\]
This ODE is linear. Compute the integrating factor:
\[\ln\mu(x)=\int\frac{2x+1}{x}\,\mathrm{d}x=\int\left(2+\frac{1}{x}\right)\,\mathrm{d}x=2x+\ln x\]which means the integrating factor \(\mu(x)=e^{2x+\ln x}=xe^{2x}\).

Multiply both sides by \(\mu\) and you can solve for \(y\):
\[\begin{align*}xe^{2x}\frac{\mathrm{d}y}{\mathrm{d}x}+e^{2x}(2x+1)y&=x\\[1ex]
\frac{\mathrm{d}}{\mathrm{d}x}\left[xe^{2x}y\right]&=x
\end{align*}\]Integrate both sides, etc. I'll leave the rest for you.