Question

By eliminating the arbitrary constants A and B from $y=A{x}^2+Bx$, we get the differential equation :

• A. $\frac{d^{3}y}{d{x}^3}=0$
• B. $x^2\frac{d^{2}y}{d{x}^2}-2x\frac{dy}{dx}+2y=0$
• C. $\frac{d^{2}y}{d{x}^2}=0$
• D. $x^2\frac{d^{2}y}{d{x}^2}+y=0$

Answer 1

Answer: (B)

$x^2\frac{d^{2}y}{d{x}^2}-2x\frac{dy}{dx}+2y=0$

$y=A{x}^2+Bx$.......(1)

$\Rightarrow y^{\prime }=2Ax+B$.........(2)

and $y^{\prime \prime }=2A\Rightarrow A=\frac{y^{\prime \prime }}{2}$

Substitute $A$ in (2) $\Rightarrow B=y^{\prime }-y^{\prime \prime }x$

Now substitute both A and B in (1)

$\Rightarrow 2y=y^{\prime \prime }{x}^2+2(y^{\prime }-y^{\prime \prime }x)x=2y^{\prime }x-y^{\prime \prime }{x}^2$

$\Rightarrow x^2\frac{d^{2}y}{d{x}^2}-2x\frac{dy}{dx}+2y=0$