7.2 Nonhomogeneous linear equations  (Page 4/9)

Variation of parameters

Sometimes, $r(x)$ is not a combination of polynomials, exponentials, or sines and cosines. When this is the case, the method of undetermined coefficients does not work, and we have to use another approach to find a particular solution to the differential equation. We use an approach called the method of variation of parameters    .

To simplify our calculations a little, we are going to divide the differential equation through by $a,$ so we have a leading coefficient of 1. Then the differential equation has the form

where $p$ and $q$ are constants.

If the general solution to the complementary equation is given by $c_1{y}_1(x)+c_2{y}_2(x),$ we are going to look for a particular solution of the form $y_p(x)=u(x)y_1(x)+v(x)y_2(x\text{).}$ In this case, we use the two linearly independent solutions to the complementary equation to form our particular solution. However, we are assuming the coefficients are functions of x , rather than constants. We want to find functions $u(x)$ and $v(x)$ such that $y_p(x)$ satisfies the differential equation. We have

Substituting into the differential equation, we obtain

Note that $y_1$ and $y_2$ are solutions to the complementary equation, so the first two terms are zero. Thus, we have

If we simplify this equation by imposing the additional condition $u^{\prime }{y}_1+v^{\prime }{y}_2=0,$ the first two terms are zero, and this reduces to $u^{\prime }{y}_1{}^{\prime }+v^{\prime }{y}_2{}^{\prime }=r(x\text{).}$ So, with this additional condition, we have a system of two equations in two unknowns:

Solving this system gives us $u^{\prime }$ and $v^{\prime },$ which we can integrate to find u and v .

Then, $y_p(x)=u(x)y_1(x)+v(x)y_2(x)$ is a particular solution to the differential equation. Solving this system of equations is sometimes challenging, so let’s take this opportunity to review Cramer’s rule, which allows us to solve the system of equations using determinants.