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

Initial Value Problems [IVPs]

To solve IVPs:

1. Find $\displaystyle y_h(t)$
2. Find $\displaystyle y_p(t)$
3. Find coefficients for $\displaystyle y_p(t)$
4. Use $\displaystyle y_g(t)=y_h(t)+y_p(t)$ to find coefficients in $\displaystyle y_h(t)$.

Example 02: Solve $\displaystyle t^2 y^{\,\prime\prime}-ty^{\,\prime}+3y=0$, $y(1)=2$ , $y^{\,\prime}(1)=1$

Solution:

From Example 01, the homogeneous solution is $\displaystyle y_h(t)=c_1 t^{1}\cos{(\sqrt{2} \ln{(t)})}+c_2 t^{1}\sin{(\sqrt{2} \ln{(t)})}$.

$y_h(1)=c_1 1^{1}\cos{(\sqrt{2} \ln{(1)})}+c_2 1^{1}\sin{(\sqrt{2} \ln{(1)})}=2$

$\Rightarrow c_1 \cos{(\sqrt{2}(0))}+\sin{(\sqrt{2} (0))}=2$

$\Rightarrow c_1 \cos{(0)}+c_2 \sin{(0)}=2$

$\Rightarrow c_1 (1)+c_2 (0)=2$

$\Rightarrow {c_1 = 2}$

and

$\displaystyle y_h^{\,\prime}(t)=c_1 \cos{(\sqrt{2} \ln{(t)})} - c1t\sin{(\sqrt{2} \ln{(t)})}\left(\frac{\sqrt{2}}{t}\right)+c_2 \sin{(\sqrt{2} \ln{(t)})} + c_2 t\cos{(\sqrt{2} \ln{(t)})}\left(\frac{\sqrt{2}}{t}\right)$

So,

$y_h^{\,\prime}(1)=c_1 \cos{(0)} - c1(1)\sin{(0)}\left(\frac{\sqrt{2}}{1}\right)+c_2 \sin{(0)} + c_2 (1)\cos{(0)}\left(\frac{\sqrt{2}}{1}\right) = 1$

$\Rightarrow c_1(1) - c1(0) +c_2 (0) + c_2 (1)\sqrt{2} = 1$

$\Rightarrow c_1 + c_2 \sqrt{2} = 1$

$\Rightarrow 2+c_2 \sqrt{2} = 1$

$\Rightarrow c_2 =-\frac{1}{\sqrt{2}}$

Thus, $\displaystyle y_h(t)=2 t^{1}\cos{(\sqrt{2} \ln{(t)})}-\frac{1}{\sqrt{2}} t^{1}\sin{(\sqrt{2} \ln{(t)})}$.

Investigation 07

Solve the following IVPs:

1. $\displaystyle t^2 y^{\,\prime\prime}-4ty^{\,\prime}+4y=0$, $y(1)=-2$, $y^{\,\prime}(1)=-11$.
2. $\displaystyle t^2 y^{\,\prime\prime}+2ty^{\,\prime}-30y=0$, $y(1)=3$, $y^{\,\prime}(1)=4$.
3. $\displaystyle 4t^2 y^{\,\prime\prime}+17y=0$, $y(1)=-1$, $y^{\,\prime}(1)=-\frac{1}{2}$.

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Last Modified: Wednesday, 14 October 2020 11:47 EDT