Examples

Variation of parameters - double repeated roots

Variation of parameters: double repeated roots

Suppose $y^{\prime \prime }+b{y}^{\prime }+cy=0$ is an ODE with $b^2=4c$ and therefore $r^2+br+c=(r+\frac{b}{2}{)}^2$ and $r=-\frac{b}{2}$ is a repeated root. We have seen that $e^{rt}$ is one solution. Let us change the parameter $c$ in the family $c{e}^{rt}$ to a function $v(t)e^{rt}$ and plug it in:

$$\begin{array}{rl}& (v{e}^{rt}{)}^{\prime \prime }+b(v{e}^{rt}{)}^{\prime }+cv{e}^{rt}\\ & =v^{\prime \prime }{e}^{rt}+2r{v}^{\prime }{e}^{rt}+r^{2}v{e}^{rt}+b{v}^{\prime }{e}^{rt}+brv{e}^{rt}+cv{e}^{rt}\\ & =v(r^2+br+c)e^{rt}+(v^{\prime \prime }+(2r+b)v^{\prime })e^{rt}\\ & =0+(v^{\prime \prime }+0)e^{rt}.\end{array}$$

This quantity is zero when $v^{\prime \prime }=0$, which happens when $v^{\prime }(t)=c_2$ and $v(t)=c_{2}t+c_2^{\prime }$. It is enough to consider the specific case $v(t)=t$ because the other parameters are already accounted for when we take the span to obtain the complete set of solutions:

$$C_1{e}^{rt}+C_{2}t{e}^{rt}.$$

Variation of parameters - second nontrivial solution from trivial guess

Variation of parameters: second nontrivial solution from trivial guess

Problem: Find a complete family of solutions to the ODE given by $t{y}^{\prime \prime }+3y^{\prime }=0$.

Solution: This ODE is reducible to first order because $y$ does not appear. However, let us use variation of parameters to solve it. Write it as $y^{\prime \prime }+\frac{3}{t}{y}^{\prime }=0$. Notice first that $y_1(t)=1$ is a (rather trivial) solution. Look for a second solution of the form $v(t)\cdot 1$. Using the formula, we have:

$$v(t)=\int 1^{-2}{e}^{-\int \frac{3}{t}dt}dt=\int e^{-3\ln t}dt=\frac{t^{-2}}{-2}.$$

So $y_2(t)=t^{-2}$ is a second solution, and the complete family is given by $C_1+C_2{t}^{-2}$.

Solving an inhomogeneous equation

Solving an inhomogeneous equation

Problem: Find a complete set of solutions to the ODE given by $y^{\prime \prime }+y=\csc t$.

Solution: The corresponding homogeneous ODE is $y^{\prime \prime }+y=0$, which has characteristic equation $(r+i)(r-i)$, and its general solution is therefore $y(t)=C_1\sin t+C_2\cos t$.

To find a particular solution to the inhomogeneous, we consider varying parameters. Let $y_1=\sin t$ and $y_2=\cos t$. Using the formulas above for $v_1$ and $v_2$, we compute:

$$\begin{array}{rl}{v}_1(t)& =\int \frac{\cos t\cdot \csc t}{{\cos }^{2}t+{\sin }^{2}t}dt=\int \cot tdt=\ln |\sin t|,\\ v_2(t)& =\int -\frac{\sin t\cdot \csc t}{{\cos }^{2}t+{\sin }^{2}t}dt=\int -1dt=-t.\end{array}$$

(Notice, again, that we will not need the integration constants, since those will lead to repeat part of the homogeneous solution.)

Adding the particular inhomogeneous solution to the general homogeneous solution, we find the complete family:

$$C_1\sin t+C_2\cos t+\ln |\sin t|\cdot \sin t-t\cdot \cos t.$$