Variable Coefficient Equations

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 4.7 – Variable Coefficient Equations

Expected Educational Results

• Objective 14–1: I can identify variable-coefficient equations.
• Objective 14–2: I can find a solution to a homogeneous Cauchy-Euler equation.
• Objective 14–3: I can find a solution to a nonhomogeneous Cauchy-Euler equation.
• Objective 14–4: Given a solution, I can apply the Reduction of Order Theorem to find another solution to a Cauchy-Euler equation.
• Objective 14-5: I can find a solution to a Cauchy-Euler IVP.

Variable Coefficient Equations

Reduction of Order

Theorem: Reduction of Order

Let $y_1(t)\ne 0$ be a solution to the homogeneous DE on some interval. Then, $y_2(t)=y_1(t)\int{\frac{e^{-\int{p(t)\,dt}}}{y_1^{\,2}(t)}\,dt}$, where $p(t)$ is the coefficient of the first-order derivative term, is a second, linearly independent solution.

Example 03:

Consider $\displaystyle y^{\,\prime\prime}-ty^{\,\prime}+y=0$. Since the second-order derivative does not have a coefficient with $t^2$, this is not a Cauchy-Euler equation.

Verify $\displaystyle y_1(t)=t$ is a solution. Find $\displaystyle y_2(t)$.

Solution:

$\displaystyle y_1(t)=t$

$\displaystyle y_1^{\,\prime}(t)=1$

$\displaystyle y_1^{\,\prime\prime}(t)=0$

So, $\displaystyle y_1^{\,\prime\prime}-ty_1^{\,\prime}+y_1=0-t(1)+t=0$. Thus, $\displaystyle y_1(t)=t$ is a solution to the ode.

Since $\displaystyle y^{\,\prime\prime}-ty^{\,\prime}+y=0\Rightarrow y^{\,\prime\prime}+p(t)y^{\,\prime}+y=0 \Rightarrow p(t)=-t$.

Let's find $y_2(t)$. By the Reduction of Order definition:

$y_2(t)=y_1(t)\int{\frac{e^{-\int{p(t)\,dt}}}{y_1^{\,2}(t)}\,dt}$

$\Rightarrow y_2(t)=t\int{\frac{e^{-\int{-t\,dt}}}{t^2}\,dt}$

$\Rightarrow y_2(t)=t\int{\frac{e^{\int{t\,dt}}}{t^2}\,dt}$

$\Rightarrow y_2(t)=t\int{\frac{e^{\frac{t^2}{2}}}{t^2}\,dt}$

Thus, $\displaystyle y_h(t)=c_1 t+c_2 t\int_a^t{\frac{e^{\frac{u^2}{2}}}{u^2}\,du}$.

Method of Solving Nonhomogeneous Cauchy-Euler Equations

There are two options:

• Use method of variation of parameters; or
• Find one solution and apply the Reduction of Order Theorem

Investigation 08

Solve the following nonhomogeneous DEs:

1. $\displaystyle t^2 y^{\,\prime\prime}-4ty^{\,\prime}+4y=t^2$, $t>0$ [Hint: Use a substitution $x=\ln{(t)}$.]
2. $\displaystyle t^2 y^{\,\prime\prime}+ty^{\,\prime}+9y=-\tan{\left(3\ln{(t)}\right)}$, $t>0$.

Investigation 09

Solve the following homogeneous DEs:

1. Suppose $y_1(t)=t^2$ is a solution to $\displaystyle t^2 y^{\,\prime\prime}-3ty^{\,\prime}+4y=0$, $t>0$. Find the general solution.
2. Suppose $y_1(t)=t\sin{(\ln{(t)})}$ is a solution to $\displaystyle t^2 y^{\,\prime\prime}-ty^{\,\prime}+2y=0$, $t>0$. Find the general solution.

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Last Modified: Wednesday, 14 October 2020 2:48 EDT