Answer to Question #246938 in Differential Equations for Jac

Question

Solve the differential equation by substitution suggested by equation. Show complete solution.

(dy/dx) = 2y/x + cos(y/x²)

Answer 1

$\frac{dy}{dx}=\frac{2y}{x}+cos(\frac{y}{x^{2}})$ this is equation (i)

Let y=Vx²

Then $\frac{dy}{dx}=2Vx+x^{2}\frac{dV}{dx}$

Substituting $\frac{dy}{dx}$ in equation (i)

$2Vx+x^{2}\frac{dV}{dx}=2\frac{Vx^{2}}{x}+cos(\frac{Vx^{2}}{x^{2}})$

$x^{2}\frac{dV}{dx}=2Vx-2Vx+cos(V)$

$x^{2}\frac{dV}{dx}=cos(V)$

$\frac{dV}{cos(V)}=\frac{dx}{x^{2}}$

Integrating both sides;

$\int sec(V)dV=\int \frac{dx}{x^{2}}$

$ln(sec(V)+tan(V))=\frac{-1}{x}+C$ this is equation (ii)

Substituting $V=\frac{y}{x^{2}}$ into equation (ii)

$ln(sec(\frac{y}{x^{2}})+tan(\frac{y}{x^{2}}))=\frac{-1}{x}+C$