Ordinary Differential Equations/Exact 4

$\begin{array}{cc}& \text{Q1 answer:}\\ & \frac{dy}{dx}+2y=x^2{e}^{-2x}+5\\ & y=\frac{\int e^{\int P\left(x\right)dx}Q\left(x\right)dx+C}{e^{\int P\left(x\right)dx}}\\ & e^{\int P\left(x\right)dx}\\ & P\left(x\right)=2\\ & e^{\int 2dx}=e^{2x}\\ & Q\left(x\right)=x^2{e}^{-2x}+5\\ & y=\frac{\int \left(e^{2x}\right)(x^2{e}^{-2x}+5)dx+C}{e^{2x}}\\ & y=\frac{\int x^2+5e^{2x}dx+C}{e^{2x}}=\frac{\frac{x^3}{3}+\frac{5e^{2x}}{2}+C}{e^{2x}}\\ & \underset{\_}{y=\frac{x^3}{3e^{2x}}+\frac{5}{2}+\frac{C}{e^{2x}}}\end{array}$ Template:BookCat