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

In summary, reducing the order of an ODE with Ansatz y2 involves using a substitution method to simplify the equation and make it easier to solve. This method is most effective for linear, homogeneous equations with constant coefficients, and may not work for nonlinear or non-homogeneous equations. It is important to consider the limitations of this method before applying it to an ODE.
  • #1
samleemc
9
0
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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  • #2
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:​
 

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

How does reducing the order of ODE Ly with Ansatz y2 work?

Reducing the order of an ODE (ordinary differential equation) with Ansatz y2 involves using a substitution method to transform the original equation into a simpler form that can be solved more easily. This is done by introducing a new variable, u, and using it to replace the original function y. The resulting equation is then solved for u, and the solution is used to find the solution for the original function y.

What is the purpose of using Ansatz y2 to reduce the order of an ODE?

The purpose of using Ansatz y2 is to simplify the ODE and make it easier to solve. By introducing a new variable and rewriting the equation in terms of this variable, the original equation can often be reduced to a first-order equation, which is easier to solve than a second-order equation. This method can also help to identify patterns and relationships within the equation.

Can any ODE be reduced using Ansatz y2?

No, not all ODEs can be reduced using Ansatz y2. This method is most effective for linear, homogeneous equations with constant coefficients. Nonlinear or non-homogeneous equations may require different techniques to reduce the order.

What is the difference between reducing the order of an ODE and solving it?

Reducing the order of an ODE involves simplifying the equation and rewriting it in a different form. This allows for easier solution of the equation, but it does not provide the final solution. After reducing the order, the equation still needs to be solved for the original function y using the substitution for u.

Are there any limitations to using Ansatz y2 to reduce the order of an ODE?

Yes, there are limitations to this method. As mentioned before, not all ODEs can be reduced using Ansatz y2. Additionally, this method may not work for equations with complex solutions or equations with varying coefficients. It is important to consider the specific characteristics of the ODE before deciding to use this method to reduce its order.

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