Question

Find a constant coefficient linear second-order differential equation whose general solution is:

(c1 + c2(x))e^(-2x) + 1

Answer 1

note:a second order differential equation ay"+by'+cy=f(t) has homogeneous part ay"+by'+cy=0 .then the homogeneous solution has auxiliary equation am²+bm+c=0.if this has repeated roots m₁.then the homogeneous solution

y_h=c₁e^m₁x+c₂xe^-m₁x.general solution is y=y_h+y_p.

Answer) given y=c₁e^-2x+c₂xe^-2x+1.so the differential equation has homogeneous solution c₁e^-2x+c₂xe^-2x.from it is clear that here the auxiliary equation has repeated root -2.so auxiliary equation of the homogeneous part of the differential equation that we want to find is (m+2)²=m²+4m+4.from these we can find the homogeneous part of the differential equation is y"+4y'+4y=0.so generally the differential equation is y"+4y'+4y=f(x).by finding y,y',y" and substituting in y"+4y'+4y=f(x),we get f(x)=4

So the second order linear differential equation with solution y is

y"+4y'+4y=4.

Answer)y"+4y'+4y=4