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05 Reduction of Order

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

General Form of 2nd order DEΒΆ

F(x,y,yβ€²,yβ€²β€²)=0 F(x, y, y', y'') = 0

This is for variable coefficients.

SolvingΒΆ

  • 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 yy Independent variable xx
Form F(x,yβ€²,yβ€²β€²)=0F(x, y', y'') = 0 F(y,yβ€²,yβ€²β€²)=0F(y, y', y'') = 0
Let yβ€²=Pβ€…β€ŠβŸΉβ€…β€Šyβ€²β€²=Pβ€²y' = P \implies y'' = P' $y' = P \
\implies y'' = P' = \frac{dP}{dy} y' \ y''= P \left(\frac{dP}{dy}\right)$
Solve F(x,P,Pβ€²)=0F(x, P, P') = 0 F(y,P,PdPdy)=0F(y, P, P \frac{dP}{dy}) = 0
Substitute yβ€²=Pβ€…β€ŠβŸΉβ€…β€Šyβ€²β€²=Pβ€²y' = P \implies y'' = P' yβ€²=Pβ€…β€ŠβŸΉβ€…β€Šyβ€²β€²=Pβ€²y' = P \implies y'' = P'
Solve F(x,y)F(x, y) F(x,y)F(x, y)

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