Question

If y = a + bx², a, b arbitrary constants, then

(a) $\frac{d^{2}y}{d{x}^2}=2xy$
(b) $x\frac{d^{2}y}{d{x}^2}=y_1$
(c) $x\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}+y=0$
(d) $x\frac{d^{2}y}{d{x}^2}=2xy$

Answer 1

(b) $x\frac{d^{2}y}{d{x}^2}=y_1$

Here,
$$\begin{gathered} y=a+b{x}^2 \\ \Rightarrow y_1=2bx \\ \Rightarrow y_2=2b \\ \mathrm{Multiplying}\mathrm{by}x\mathrm{on}\mathrm{both}\mathrm{sides}\mathrm{we}\mathrm{get}, \\ x{y}_2=2bx=y_1 \\ \Rightarrow x\frac{d^{2}y}{d{x}^2}=y_1 \end{gathered}$$