Question

Which of the following differential equations has y=x as one of its particular solution?
(a) $\frac{d^{2}y}{d{x}^2}-x^2\frac{dy}{dx}+xy=x$
(b) $\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+xy=x$
(c) $\frac{d^{2}y}{d{x}^2}-x^2\frac{dy}{dx}+xy=0$
(d) $\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+xy=0$

Answer 1

Given solution is y=x ......(i)
On differentiating w.r.t. x, we get
$\frac{dy}{dx}=1$ .......(ii)
Again, differentiating w.r.t. x, we get
$\frac{d^{2}y}{d{x}^2}=0$ ......(iii)
Now, on substituting the values from Eqs.(i), (iii) and (ii) in each of the given option, we find that only the differential equation given in option (c) is satisfied by the values of y, $\frac{dy}{dx}$ and $\frac{d^{2}y}{d{x}^2}.$
$\frac{d^{2}y}{dx}-x^2\frac{dy}{dx}+xy=0-x^2.1+x.x=-x^2+x^2=0$ Hence, the option (c) is correct.