Question

dtermine the general solution...

Answer 1

of the following differential equation:
\[y ^{4} + 2 y \prime \prime \prime + 2y \prime \prime = 3e ^{x} + 2xe ^{-x} + e ^{-x}sinx\]

Answer 2

@electrokid @amistre64

Answer 3

@abb0t @ASAAD123 @Callisto

Answer 4

there is also a table method solution ...

Answer 5

but for this case, i might just try the homogenous solution to get the y1, y2, ....

Answer 6

r^4+2r^3+2r^2=0

r^2(r^2+2r+2)=0 , so we have 2 muliples of r=0

and

r = (-2+- sqrt(4-4(2)))/2
r = (-2+- sqrt(-4))/2
r = (-2+- 2i)/2

r = (-1+ i) , (-1- i)

Answer 7

\[y_h=c_0e^{0x}+c_1xe^{0x}+c_3e^{-1+i}+c_4e^{-1-i}\]
\[y_h=c_0+c_1x+c_3e^{-1}e^{i}+c_4e^{-1}e^{-i}\]