05 Reduction of Order
General Form of 2nd order DE¶
\[ F(x, y, y', y'') = 0 \]
This is for variable coefficients.
Solving¶
- 2nd order DE is reduced into two 1st order DE
- they are solved one after each other
Reduction of order method is possible under 2 cases
| Case 1 | Case 2 | |
|---|---|---|
| missing terms | Dependent variable \(y\) | Independent variable \(x\) |
| Form | \(F(x, y', y'') = 0\) | \(F(y, y', y'') = 0\) |
| Let | \(y' = P \implies y'' = P'\) | $y' = P \ |
| \implies y'' = P' = \frac{dP}{dy} y' \ y''= P \left(\frac{dP}{dy}\right)$ | ||
| Solve | \(F(x, P, P') = 0\) | \(F(y, P, P \frac{dP}{dy}) = 0\) |
| Substitute | \(y' = P \implies y'' = P'\) | \(y' = P \implies y'' = P'\) |
| Solve | \(F(x, y)\) | \(F(x, y)\) |