The ODE is linear. Multiplying both sides by 
allows you to condense the left hand side as the derivative of a product.
![\dfrac{\mathrm d}{\mathrm dt}\left[e^{7t}x\right]=5e^{7t}\cos2t](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dt%7D%5Cleft%5Be%5E%7B7t%7Dx%5Cright%5D%3D5e%5E%7B7t%7D%5Ccos2t)
Integrating both sides with respect to
yields
The integral can be done by parts. You should get
With the given initial condition, you have
So the particular solution to the ODE is
Integrating both sides with respect to
The integral can be done by parts. You should get
With the given initial condition, you have
So the particular solution to the ODE is