- #1
Abraham
- 69
- 0
Homework Statement
Find the general solution of
[tex]\frac{dy}{dt}[/tex] = 2y +sin(2t)
Homework Equations
The general solution of a nonhomogeneous ode is the particular solution of the nonhomo plus the solution of the homogeneous ode.: y(t)= y[tex]_{p}[/tex](t)+y[tex]_{h}[/tex](t)
The Attempt at a Solution
[tex]\frac{dy}{dt}[/tex] - 2y = sin(2t)
Initial guess, is that y[tex]_{p}[/tex] = a*Sin(2t)+b*Cos(2t), where a and b are unknown constants later to be found
Then, the d.e. becomes:
[tex]\frac{d}{dt}[/tex] ( a*Sin(2t)+b*Cos(2t) ) - 2( a*Sin(2t)+b*Cos(2t) ) = Sin2t
=2aCos(2t)-2bSin(2t) - 2aSin(2t)+2bCos(2t) = Sin2t
=(2a+2b)*Cos(2t) + (-2b-2a)*Sin(2t) = Sin2t
Using the fact that polynomial coefficients must be equal:
2a+2b=0
-2b-2a=1
And here is my issue. These simultaneous equations are false, and thus I cannot possibly solve the d.e.
Can someone tell me where I messed up? Thanks
PS. I know this can also be found using integrating factors, but i need help on the section with "guessing" the solution