Question

Solve
d^2y/dx^2 + 1/x×dy/dx= 12Inx/x^2

Answer 1

Let us solve the equation $\frac{d^2y}{dx^2}+\frac{1}{x}\frac{dy}{dx}=12\frac{\ln x}{x^2}$.

Let us multiply both part by $x$: $x\frac{d^2y}{dx^2}+\frac{dy}{dx}=12\frac{\ln x}{x}$. Then

$\frac{d}{dx}(x\frac{dy}{dx})=12\frac{\ln x}{x}$ and therefore,

$x\frac{dy}{dx}=12\int \frac{\ln x}{x}dx=12\int \ln x\ d(\ln x)=6\ln^2x+C_1$

Let us divide both part by $x$:

$\frac{dy}{dx}=6\frac{\ln^2x}{x}+\frac{C_1}{x}$

Consequently,

$y=6\int(\frac{\ln^2x}{x}+\frac{C_1}{x})dx=6\int\ln^2x\ d(\ln x)+6C_1\int\frac{dx}{x}=2\ln^3 x+6C_1\ln x+C_2$

Answer: $y=2\ln^3 x+6C_1\ln x+C_2$