Question

x2(d2y/dx2) - 3x(dy/dx)+5y = xlogx​

Answer 1

Answer:

x2y′′−3xy′+5y=x2sin(logx).

Since it is second order linear differential equation (Euler - Cauchy equation), the homogeneous part of equation can be solved as follows,

Let y=xr, to get the characteristic equation, which is

r(r−1)−3r+5=r2−4r+5=0

r=2+i,2−i.

Thus the homogeneous solution yh=c1x2sin(logx)+c2x2cos(logx).

Now the particular solution is found by many ways, here I will just guess the particular solution to be ax2ln(x)cos(logx).

To find the value of the constant a, we substitute the the particular solution into differential equation.

ax2((logx+3)cos(logx)−(3logx+2)sin(logx))+ax2((2log(x)+1)cos(logx)−logxsin(logx))−4ax2lnxcos(logx)=−2ax2sin(logx)=x2sin(logx)

So a=−12.

Thus the solution is y=c1x2sin(logx)+c2x2cos(logx)−12x2ln(x)cos(logx).

Answer 2

Step-by-step explanation:

x2(d2y/dx2) - 3x(dy/dx)+5y = xlogx