Is c1 + c2 x^.5 a Solution of y y'' + (y')^.5 = 0?

In summary, the conversation discusses verifying that given functions are solutions to a differential equation and determining if a specific form of solution is valid. The participants also briefly discuss the importance of linearity in determining the validity of fundamental solutions for a differential equation.
  • #1
roeb
107
1

Homework Statement



Verify that y1(x) = 1 and y2(x) = x^.5 are solutions of the following y y'' + (y')^.5 = 0. Then show that c1 + c2 x^.5 is not in general a solution of this equation.

Homework Equations





The Attempt at a Solution



I was able to show that both y1 and y2 are solutions to the DE. I found the Wronskian to be 1/(2 sqrt(x)) which is not equal to zero, so I was under the impression that this would mean that the two solutions would form a fundamental set of solution. Does anyone see why c1 + c2 x^.5 isn't a solution?
 
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  • #2
Your ODE isn't linear.
 
  • #3
Does it matter if it's not linear in general? Boyce/Diprima's theorems don't seem to make note of whether or not the ODE must be linear for a set of fundamental solutions to be valid.
 
  • #4
I'd have to see the theorem, but linearity is the property that tells you if y1 and y2 are solutions to an ODE then so is c1*y1+c2*y2. I think you need it.
 

FAQ: Is c1 + c2 x^.5 a Solution of y y'' + (y')^.5 = 0?

What is a fundamental solution for an ODE?

A fundamental solution for an ODE (ordinary differential equation) is a solution that satisfies the equation and is independent of any initial or boundary conditions. It is the building block for finding specific solutions to the ODE when combined with the initial or boundary conditions.

How is a fundamental solution different from a particular solution?

A fundamental solution is a general solution that satisfies the ODE, while a particular solution is a specific solution that satisfies the ODE with given initial or boundary conditions. A fundamental solution can be used to find a particular solution by incorporating the given conditions.

Can a fundamental solution have multiple solutions?

No, a fundamental solution is unique for a given ODE. However, it can be used to generate an infinite number of particular solutions by incorporating different initial or boundary conditions.

How is a fundamental solution related to the homogenous solution of an ODE?

A fundamental solution is a special type of homogenous solution that is independent of any initial or boundary conditions. It serves as a basis for finding other homogenous solutions by incorporating the initial or boundary conditions.

In what cases is a fundamental solution not applicable?

A fundamental solution is not applicable for non-linear ODEs or when the ODE has discontinuous coefficients. It also cannot be used if the ODE has singularities or if the initial or boundary conditions are not well-defined.

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