Non Homogeneous Linear equations
$$y''+4y=e^{3x}$$
I understand that the auxiliary equation is
$$r^2+4=0$$
The roots are $$\pm 2$$
but this is about as for as I understand the process

Question

Non Homogeneous Linear equations
$$y''+4y=e^{3x}$$
I understand that the auxiliary equation is 
$$r^2+4=0$$
The roots are $$\pm 2$$
but this is about as for as I understand the process

anonymous · Answer

Why do we have sines and cosines or is that just part of the process?

anonymous · Answer

+/- 2i

anonymous · Answer

which means use a linear combination of sin and cos for the for of your solution

anonymous · Answer

constants go out front:                         C1*sin (  )
eigenvalues are the angular freq:          C1*sin(2*t)

anonymous · Answer

http://tutorial.math.lamar.edu/Classes/DE/ComplexRoots.aspx

anonymous · Answer

whoops I left a bit out   solutions should actually look like :
 C1*cos(2*t) + C2*i*sin(2*t)

(I left out the 'i')

anonymous · Answer

that help?

anonymous · Answer

oh I remember now

anonymous · Answer

to get a particular solution of the non homogeneous equation
$$
y= A e ^{3 x}\
y'= 3 A e^{3 x}\
y''=9 A e^{3 x}\
y''(x) + 4 y[x]= 13 Ae^{3 x}=e^{3 x}\
A=\frac 1 {13}
y(x)=\frac 1 {13}e^{3 x}

$$
The general solution is
$$
y= C_1 \cos(2x) + c_2 \sin(2x) +\frac 1 {13}e^{3 x}
$$