Question

Find the homogeneous equation with constant coefficients that has the given general solution: http://i42.tinypic.com/2m7x8aa.jpg

Answer 1

\[y=C_1(e){}^{4 x}+C_2(x)+C_3\left(+C_4(e){}^x\right) \cos (2 x)+C_5(e){}^x \sin (2 x)\]

Answer 2

\[\tiny y=2 x^2+(2 e)^{4 x}+\left(\frac{2}{5} \left(5 e-20 e^3+16 e^5\right)\right)^x \sin (2 x)+\frac{2}{3} \left(2^{2-3 x} \left(1-8 e^2+8 e^4\right)^{3 x}-3 \left(\frac{1}{2} \left(1-8 e^2+8 e^4\right)\right)^x\right) \cos (2 x)-1\]