Question

The respective expressions for complimentary function and particular integral part of the solution of the differential equation $\frac{d^{4}y}{d{x}^4}+3\frac{d^{2}y}{d{x}^2}=108x^2$ are

• A. $[c_1+c_{2}x+c_{3}sin\sqrt{3}x+c_{4}cos\sqrt{3}x]$ and $[3x^4-12x^2+c]$
• B. $[c_{2}x+c_{3}sin\sqrt{3}x+c_{4}cos\sqrt{3}x]$ and $[5x^4-12x^2+c]$
• C. $[c_1+c_{3}sin\sqrt{3}x+c_{4}cos\sqrt{3}x]$ and $[3x^4-12x^2+c]$
• D. $[c_1+c_{2}x+c_{3}sin\sqrt{3}x+c_{4}cos\sqrt{3}x]$ and $[5x^4-12x^2+c]$

Answer 1

The correct option is A $[c_1+c_{2}x+c_{3}sin\sqrt{3}x+c_{4}cos\sqrt{3}x]$ and $[3x^4-12x^2+c]$

$\frac{d^{4}y}{d{x}^4}+3\frac{d^{2}y}{d{x}^2}=108x^2$
For complementary solution
$D^4+3D=0$
$D^2(D^2+3)=0$
$D^2=0$
$D^2+3=0$ $\Rightarrow$ $D=\pm i\sqrt{3}$
So, complementary solution:
C.F.$=C_1+C_{2}x+C_{3}sin\sqrt{3}x+C_{4}cos\sqrt{3}x$
Particular integral
$=\frac{108x^2}{D^4+3D^2}=\frac{1}{3D^2}\left[\frac{1}{1+\frac{D^2}{3}}\right]\left(108x^2\right)$
P.I. = $3x^4-12x^2+C$