Question

what is the solution for this Linear Differential equation (x^2D^2-3xD+5)y =x^2(logx) ​

Answer 1

Answer:

Since this is a Cauchy-Euler differential equation, we make the change of (dependent) variable x=et . This can be rewritten as t=lnx .

By the Chain Rule,

dydx=dydtdtdx=1xdydt, and

d2ydx2=ddx(1xdydt)=−1x2⋅dydt+1x⋅1xd2ydt2.

Substituting this into the original differential equation yields

x2⋅(−1x2dydt+1x2d2ydt2)−3x⋅1xdydt−5y=(et)2⋅sint,

which reduces to

d2ydt2−4dydt−5y=e2tsint.

This transformed differential equation, now having constant coefficients, has characteristic equation r2−4r−5=(r+1)(r−5)=0 which has roots t=−1,5 .

Hence, a homogeneous solution is given by

yh=C1e−t+C2e5t.

Next, for a particular solution, we assume that yp=e2t(Acost+Bsint) .

Differentiating and substituting into the differential equation yields

e2t((3A+4B)cost+(−4A+3B)sint)−4⋅e2t((2A+B)cost+(2B−A)sint)−5⋅e2t(Acost+Bsint)=e2tsint.

This simplifies to

−10Acost−10Bsint=sint.

Equating like coefficients gives us A=0 and B=−110 .

Thus, the general solution (in t ) is given by

y=C1e−t+C2e5t−110e2tsint.

Finally since t=lnx , a general solution to the original differential equation is

y=C1x−1+C2x5−110x2sin(lnx).