Q48 E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Complete the proof of Theorem  8 by solving equation (16).

Step-by-Step Solution

Answer

The proof of Theorem 8 is ${\mathrm{y}}_2={\mathrm{vy}}_1={\mathrm{y}}_1\int \frac{-{\mathrm{e}}^{\int \mathrm{p}\left(\mathrm{t}\right)\mathrm{dt}}}{{\mathrm{y}}_1^2}\mathrm{dt}$ by solving equation (16).

Step 1: Using the Equation

From (16) we have ${\mathrm{y}}_1\mathrm{w}\text{'}+\left({2\mathrm{y}}_1^{}\text{'}+{\mathrm{py}}_1\right)\mathrm{w}=0$then

 ${\mathrm{y}}_1\mathrm{w}\text{'}=-2\left({\mathrm{y}}_1^{}\text{'}+{\mathrm{py}}_1\right)\mathrm{w}\frac{\mathrm{w}\text{'}}{\mathrm{w}}=-\frac{2\left({\mathrm{y}}_1^{}\text{'}+{\mathrm{py}}_1\right)}{{\mathrm{y}}_1}$

Integrate on both sides.

 $$\begin{gathered} \int \frac{\mathrm{w}\text{'}}{\mathrm{w}}\mathrm{dw}=-\int \frac{2\left({\mathrm{y}}_1^{}\text{'}+{\mathrm{py}}_1\right)}{{\mathrm{y}}_1}\mathrm{dt}\int \frac{\mathrm{w}\text{'}}{\mathrm{w}}\mathrm{dw} \\ =-\int \left(2\frac{{\mathrm{y}}_1^{}\text{'}}{{\mathrm{y}}_1}+\mathrm{p}\right)\mathrm{dt}-\mathrm{lnw} \\ =2{\mathrm{lny}}_1+\int \mathrm{p}\left(\mathrm{t}\right)\mathrm{dtw} \\ ={\mathrm{e}}^{-\left(2{\mathrm{lny}}_1+\int \mathrm{p}\left(\mathrm{t}\right)\mathrm{dt}\right.}\mathrm{w} \\ =\frac{-{\mathrm{e}}^{\int \mathrm{p}\left(\mathrm{t}\right)\mathrm{dt}}}{{\mathrm{y}}_1^2} \end{gathered}$$

Step 2: Substitute the value for "w"

One has $\mathrm{w}=\mathrm{v}\text{'}$,Therefore,

$$\begin{gathered} \frac{\mathrm{dv}}{\mathrm{dt}}=\frac{-{\mathrm{e}}^{\int \mathrm{p}\left(\mathrm{t}\right)\mathrm{dt}}}{{\mathrm{y}}_1^2}\mathrm{dv} \\ =\frac{-{\mathrm{e}}^{\int \mathrm{p}\left(\mathrm{t}\right)\mathrm{dt}}}{{\mathrm{y}}_1^2}\mathrm{dt} \\ \mathrm{v}=\int \frac{-{\mathrm{e}}^{\int \mathrm{p}\left(\mathrm{t}\right)\mathrm{dt}}}{{\mathrm{y}}_1^2}\mathrm{dt} \end{gathered}$$

And

${\mathrm{y}}_2={\mathrm{vy}}_1={\mathrm{y}}_1\int \frac{-{\mathrm{e}}^{\int \mathrm{p}\left(\mathrm{t}\right)\mathrm{dt}}}{{\mathrm{y}}_1^2}\mathrm{dt}$