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Solve by using the method of undetermined coefficients. π₯ 2 π 2π¦ ππ₯ 2 β 3π₯ ππ¦ ππ₯ + 5π¦ = π₯ 2 sin ππππ₯Β

Harshit Singh
6 months ago
Dear Student

Given Differential equation isπ₯2π¦″−3π₯π¦′+5π¦=π₯2π ππ(ππππ₯).
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,

Letπ¦=π₯π,y=xr,to get the characteristic equation, which is

π(π−1)−3π+5=π2−4π+5=0r(r−1)−3r+5
=r2−4r+5=0

π=2+π,2−π.r=2+i,2−i.

Thus the homogeneous solutionπ¦β=π1π₯2π ππ(ππππ₯)+π2π₯2πππ (ππππ₯).yh=c1x2sin(logx)+c2x2cos(logx).

Now the particular solution is found by many ways, here I will just guess the particular solution to beππ₯2ππ(π₯)πππ (ππππ₯).ax2ln(x)cos(logx).

To find the value of the constantπ,a,we substitute the the particular solution into differential equation.

ππ₯2((ππππ₯+3)πππ (ππππ₯)−(3ππππ₯+2)π ππ(ππππ₯))+ππ₯2((2πππ(π₯)+1)πππ (ππππ₯)−ππππ₯π ππ(ππππ₯))−4ππ₯2πππ₯πππ (ππππ₯)=−2ππ₯2π ππ(ππππ₯)=π₯2π ππ(ππππ₯)ax2((logx+3)cos(logx)−(3logx+2)sin(logx))+ax2((2log(x)+1)cos(logx)−logxsin(logx))−4ax2lnxcos(logx)=−2ax2sin(logx)=x2sin(logx)

Soπ=−12.a=−12.

Thus the solution isπ¦=π1π₯2π ππ(ππππ₯)+π2π₯2πππ (ππππ₯)−12π₯2ππ(π₯)πππ (ππππ₯).

Thanks