Question

Form the differential equation corresponding to $xy=a{x}^2+\frac{b}{x},(a,b)$.

Answer 1

$\begin{array}{c}xy=a{x}^2+\frac{b}{x},\left(a,b\right)\\ \Rightarrow x^{2}y=a{x}^3+b\\ differentiatew.r.to.x\\ \to \frac{d\left(x^{2}y\right)}{dx}=a\frac{d{x}^3}{dx}+\frac{db}{dx}\\ \Rightarrow x^2\frac{dy}{dx}+y\frac{d{x}^2}{dx}=3a{x}^2+0\\ \Rightarrow xdy/dx+2xy=3a{x}^2\\ \Rightarrow \frac{x}{x^2}dy/dx+\frac{2xy}{x^2}3a\\ \Rightarrow \frac{1}{x}dy/dx+\frac{2y}{x}=3a\\ againdifferentiatew.r.tox\\ \left[x\cdot \frac{d\left(dy/dx\right)}{dx}-dy/dx\right]\times \frac{1}{x^2}+2\left[xdy/dx-y\frac{dx}{dx}\right]=0\\ \frac{x{d}^{2}y/d{x}^2-dy/dx}{x^2}+\frac{2xdy/dx-2y}{x^2}=0\\ x{d}^{2}y/d{x}^2-dy/dx+2xdy/dx+y=0\\ x{d}^{2}y/d{x}^2+dy/dx\left(x-1\right)-2y=0\\ Itisrequireddifferentiateequations.Ans.\end{array}$