Q33E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Use Abel's formula (Problem) to determine (up to a constant multiple) the Wronskian of two solutions on $\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{\infty }\mathbf{)}$ to $\mathbf{ty}\text{'}\text{'}\mathbf{+}\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{y}\text{'}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{=}\mathbf{0}$.

Step-by-Step Solution

Answer

The solution for the given equation $\mathrm{ty}\text{'}\text{'}+(\mathrm{t}-1)\mathrm{y}\text{'}+3\mathrm{y}=0$ is $\mathrm{W}\left[{\mathrm{y}}_1,{\mathrm{y}}_2\right]={\mathrm{cte}}^{-\mathrm{t}}$.

Step 1: Divide by on both sides

Given differential equation is:

$\begin{array}{rcl}\mathrm{ty}\text{'}\text{'}+(\mathrm{t}-1)\mathrm{y}\text{'}+3\mathrm{y}& =& 0\\ \mathrm{y}\text{'}\text{'}+\frac{\mathrm{t}-1}{\mathrm{t}}\mathrm{y}\text{'}+\frac{3}{\mathrm{t}-1}\mathrm{y}& =& 0\end{array}$

Here $\mathbf{p}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\frac{\mathbf{t}\mathbf{-}\mathbf{1}}{\mathbf{t}}\mathbf{=}\mathbf{1}\mathbf{-}\frac{\mathbf{1}}{\mathbf{t}}$

Step 2: Use Abel’s formula

Then from problem number 32

$\begin{array}{rcl}\mathrm{W}\left[{\mathrm{y}}_1,{\mathrm{y}}_2\right]& =& {\mathrm{ce}}^{{\int }_{{\mathrm{t}}_0}^{\mathrm{t}}-\left(1-\frac{1}{\mathrm{t}}\right)\mathrm{dt}}\\ & =& {\mathrm{ce}}^{{\int }_{{\mathrm{t}}_0}^{\mathrm{t}}\left(\frac{1}{\mathrm{t}}-1\right)\mathrm{dt}}\\ & =& {\mathrm{ce}}^{(\mathrm{lnt}-\mathrm{t})}\\ & =& {\mathrm{ce}}^{\mathrm{lnt}}{\mathrm{e}}^{-\mathrm{t}}\\ & =& {\mathrm{cte}}^{-\mathrm{t}}\end{array}$

$\mathrm{W}\left[{\mathrm{y}}_1,{\mathrm{y}}_2\right]={\mathrm{cte}}^{-\mathrm{t}}$