Linear Differential Equations with Variable Coefficients

Q. 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

1. $[c_1+c_{2}x+c_{3}sin\sqrt{3}x+c_{4}cos\sqrt{3}x]$ and $[3x^4-12x^2+c]$
2. $[c_{2}x+c_{3}sin\sqrt{3}x+c_{4}cos\sqrt{3}x]$ and $[5x^4-12x^2+c]$
3. $[c_1+c_{3}sin\sqrt{3}x+c_{4}cos\sqrt{3}x]$ and $[3x^4-12x^2+c]$
4. $[c_1+c_{2}x+c_{3}sin\sqrt{3}x+c_{4}cos\sqrt{3}x]$ and $[5x^4-12x^2+c]$

Q. Consider the differential equation $x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}-y=0$. Which of the following is a solution to this differential equation for $x>0$ ?

1. $e^x$
2. $x^2$
3. $1/x$
4. lnx

Q. The general solution of the differential equation $x^2\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}+y=0$ is

1. $Ax+B{x}^2$ ($A,B$ are constants)
2. $Ax+Blogx$ ($A,B$ are constants)
3. $Ax+B{x}^{2}logx$ ($A,B$ are constants)
4. $Ax+Bxlogx$ ($A,B$ are constants)

Q. $f=a_0{x}^n+a_1{x}^{n-1}y+...+a_{n-1}x{y}^{n-1}+a_n{y}^n$ where $a_i(i=0ton)areconstantthenx\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=$

1. $\frac{f}{n}$
2. $\frac{n}{f}$
3. nf
4. $n\sqrt{f}$