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 \( y^{''} - 5y^{'} + 6y = 2e^t \).

Try these questions:
Take the Quiz: