Question

Find a general solution of the following differential equation

dy/dx - 5y = (xe^-2x)*y^-2

Answer 1

$\text{Solve Bernoulli's equation } \frac{d y(x)}{d x}-5 y(x)=\frac{e^{-2 x} x}{y(x)^{2}} :\\[4mm] \text{Multiply both sides by }3 y(x)^{2} :\\[2mm] 3 \frac{d y(x)}{d x} y(x)^{2}-15 y(x)^{3}=3 e^{-2 x} x\\[4mm] \text{Let } v(x)=y(x)^{3},\text{ which gives } \frac{d v(x)}{d x}=3 y(x)^{2} \frac{d y(x)}{d x}:\\ \frac{d v(x)}{d x}-15 v(x)=3 e^{-2 x} x\\[4mm] \text{ Let } \mu(x)=e^{\int-15 d x}=e^{-15 x}.\\ \text{ Multiply both sides by } \mu(x): \\[2mm] e^{-15 x} \frac{d v(x)}{d x}-\left(15 e^{-15 x}\right) v(x)=3 e^{-17 x} x\\ \text{ Substitute } -15 e^{-15 x}=\frac{d}{d x}\left(e^{-15 x}\right) :\\ e^{-15 x} \frac{d v(x)}{d x}+\frac{d}{d x}\left(e^{-15 x}\right) v(x)=3 e^{-17 x} x\\[4mm] \text{ Apply the reverse product rule } f \frac{d g}{d x}+g \frac{d f}{d x}=\frac{d}{d x}(f g) \text{ to the left-hand side: } \frac{d}{d x}\left(e^{-15 x} v(x)\right)=3 e^{-17 x} x\\$

$\text{ Evaluate the integrals:}\\[2mm] e^{-15 x} v(x)=3 e^{-17 x}\left(-\frac{x}{17}-\frac{1}{289}\right)+c_{1}, \text{ where } c_{1} \text{ is an arbitrary constant.}\\[3mm] \text{ Divide both sides by } \mu(x)=e^{-15 x} :\\[2mm] v(x)=\frac{1}{289} e^{-2 x}\left(-51 x+289 c_{1} e^{17 x}-3\right)\\[4mm] \text{ Solve for } y(x) \text{ in } v(x)=y(x)^{3}:\\[2mm] \begin{aligned} &y(x)=\frac{e^{-(2 x) / 3} \sqrt[3]{-51 x+289 c_{1} e^{17 x}-3}}{17^{2 / 3}} \\ &\text { or } y(x)=-\frac{\sqrt[3]{-1} e^{-(2 x) / 3} \sqrt[3]{-51 x+289 c_{1} e^{17 x}-3}}{17^{2 / 3}} \\ &\text { or } y(x)=\frac{(-1)^{2 / 3} e^{-(2 x) / 3} \sqrt[3]{-51 x+289 c_{1} e^{17 x}-3}}{17^{2 / 3}} \end{aligned} \\[4mm] \text{ Simplifying the arbitrary constants we have:}\\[4mm] \begin{aligned} &y(x)=\frac{e^{-(2 x) / 3} \sqrt[3]{-51 x+c_{1} e^{17 x}-3}}{17^{2 / 3}} \\ &\text { or } y(x)=-\frac{\sqrt[3]{-1} e^{-(2 x) / 3} \sqrt[3]{-51 x+c_{1} e^{17 x}-3}}{17^{2 / 3}} \\ &\text { or } y(x)=\frac{(-1)^{2 / 3} e^{-(2 x) / 3} \sqrt[3]{-51 x+c_{1} e^{17 x}-3}}{17^{2 / 3}} \end{aligned}$