# Variation of Parameters

For higher-order non-homogeneous differential equations, we can use the Variation of Parameters method which invovles solving the complementary solution $$(y_c)$$ for the homogeneous equation and then the particular solution $$(y_p)$$. The general solution is:

$$y = y_c + y_p$$

To find the particular solution, we consider a solution $$y_p = u_1 y_1 + u_2 y_2$$. Where $$y_1, y_2$$ are solutions of the complementary solution and $$u_1, u_2$$ are non-constant coefficients. Plugging this into the standard form second-order DE gives the restrictions:

$$u_1^{'} y_1 + u_2^{'} y_2 = 0$$

$$u_1^{'} y_1^{'} + u_2^{'} y_2^{'} = g(x)$$

We will use these restrictions to solve for the functions $$u_1, u_2$$. The steps to use the variation of parameter method are:

1. Solve the general complementary solution for the homogeneous DE.
2. Solve the system of equations to solve for $$u_1^{'}, u_2^{'}$$.
3. Integrate to solve for $$u_1, u_2$$.
4. Obtain the general particular solution $$y_p = u_1 y_1 + u_2 y_2$$.
5. Obtain the general solution combinding the complementary and particular soltuions $$y = y_c + y_p$$.

Solve the general solution to $$y^{''} - 5y^{'} + 6y = 2e^t$$.