s7b said:Hi,
When using the method of variation of parameters to solve something like;
y'' + y' = 2^x
I got the aux. equation: r^2 - r =0 which gives the roots r=0,1
How do I find the complementary equation yc?
The Method of Variation of Parameters is a mathematical technique used to find a particular solution to a non-homogeneous linear differential equation. It involves using a trial solution and then solving for the undetermined coefficients.
This method is used when solving non-homogenous linear differential equations that cannot be solved using other techniques such as the Method of Undetermined Coefficients or the Laplace Transform.
The first step is to find the complementary solution to the differential equation. Then, a trial solution is chosen and substituted into the equation. The coefficients of the trial solution are then determined by solving a system of equations. Finally, the particular solution is found by substituting the coefficients into the trial solution.
One advantage is that it can be used to solve a wider range of non-homogeneous differential equations compared to other techniques. It also allows for more flexibility in choosing the trial solution, making it easier to find the particular solution.
One limitation is that it can be more complicated and time-consuming compared to other methods. It also requires some knowledge and understanding of linear algebra to solve the system of equations for the coefficients. Additionally, it may not work for all types of non-homogeneous differential equations.