Q52 E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

The reduction of order formula $\left(13\right)$  can also be derived from Abel’s identity (Problem 32). Let $\mathrm{f}\left(\mathrm{t}\right)$ be a nontrivial solution to  and  a second linearly independent solution. Show that ${\left(\frac{\mathrm{y}}{\mathrm{f}}\right)}^{\text{'}}=\frac{\mathrm{W}[\mathrm{f},\mathrm{y}]}{{\mathrm{f}}^2}$and then use Abel's identity for the Wronskian $\mathrm{W}\left[\mathrm{f},\mathrm{y}\right]$  to obtain the reduction of order formula.

Step-by-Step Solution

Answer

The solution of the given equation is $\mathrm{y}=\mathrm{f}\int \frac{-{\mathrm{e}}^{\int \mathrm{p}\left(\mathrm{t}\right)\mathrm{dt}}}{{\mathrm{f}}^2}\mathrm{dt}$.

Step 1: Using the formula

From the given  ${\left(\frac{\mathrm{y}}{\mathrm{f}}\right)}^{\text{'}}=\frac{\mathrm{fy}\text{'}-\mathrm{f}\text{'}\mathrm{y}}{{\mathrm{f}}^2}$and we know $\mathrm{w}[\mathrm{f},\mathrm{y}]=\mathrm{fy}\text{'}-\mathrm{f}\text{'}\mathrm{y}$

${\left(\frac{\mathrm{y}}{\mathrm{f}}\right)}^{\text{'}}=\frac{\mathrm{W}[\mathrm{f},\mathrm{y}]}{{\mathrm{f}}^2}$

 Hence proved.

Step 2: Integration

Now integrate on both sides;

$$\begin{gathered} \int {\left(\frac{\mathrm{y}}{\mathrm{f}}\right)}^{\text{'}}\mathrm{dt}=\int \frac{\mathrm{W}[\mathrm{f},\mathrm{y}]}{{\mathrm{f}}^2}\mathrm{dt}\frac{\mathrm{y}}{\mathrm{f}} \\ =\int \frac{-{\mathrm{e}}^{\int \mathrm{p}\left(\mathrm{t}\right)\mathrm{dt}}}{{\mathrm{f}}^2}\mathrm{dt} \\ \mathrm{y}=\mathrm{f}\int \frac{-{\mathrm{e}}^{\int \mathrm{p}\left(\mathrm{t}\right)\mathrm{dt}}}{{\mathrm{f}}^2}\mathrm{dt} \end{gathered}$$

$$\begin{gathered} \int {\left(\frac{\mathrm{y}}{\mathrm{f}}\right)}^{\text{'}}\mathrm{dt}=\int \frac{\mathrm{W}[\mathrm{f},\mathrm{y}]}{{\mathrm{f}}^2}\mathrm{dt}\frac{\mathrm{y}}{\mathrm{f}} \\ =\int \frac{-{\mathrm{e}}^{\int \mathrm{p}\left(\mathrm{t}\right)\mathrm{dt}}}{{\mathrm{f}}^2}\mathrm{dt} \\ \mathrm{y}=\mathrm{f}\int \frac{-{\mathrm{e}}^{\int \mathrm{p}\left(\mathrm{t}\right)\mathrm{dt}}}{{\mathrm{f}}^2}\mathrm{dt} \end{gathered}$$