05 Reduction of Order

Last Updated: 2 years ago2023-01-25 ; Contributors: AhmedThahir

General Form of 2^nd order DE

$$F(x, y, y', y'') = 0$$

This is for variable coefficients.

Solving

• 2^nd order DE is reduced into two 1^st 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	$F(x, y)$	$F(x, y)$

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