Reducing Order of ODE Ly with Ansatz y2: Find u to Solve

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SUMMARY

The discussion focuses on reducing the order of the ordinary differential equation (ODE) Ly ≡ (x +1)²y′′− 4(x +1)y′+6y =0 using the ansatz y2(x)= u(x)(x+1)². The known solution y[1]=(x+1)² is utilized to find a second independent solution y2. Participants seek guidance on how to derive the function u and simplify the equation to solve (x+1)²u''+6u=0.

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  • Understanding of ordinary differential equations (ODEs)
  • Familiarity with the method of reduction of order
  • Knowledge of the ansatz technique in solving differential equations
  • Basic calculus, particularly differentiation and integration
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samleemc
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Ly ≡ (x +1)^2y′′− 4(x +1)y′+6y =0

given y[1]=(x+1)^2 is a solution, use the ansatz y2(x)= u(x)(x+1)2 to reduce
the order of the differential equation and find a second independent solution y2

how to reduce !? and i can't find u ...can't solve (x+1)^2u''+6u=0

please help!
thx!
 
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Welcome to PF!

Hi samleemc! Welcome to PF! :smile:

(try using the X2 and X2 tags just above the Reply box :wink:)

Put y = u(x) (x+1)2 into the original equation …

what do you get? :smile:​
 

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