Differential Equations : Variation of Parameters
Study concepts, example questions & explanations for Differential Equations
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Example Questions
Example Question #1 : Variation Of Parameters
Using Variation of Parameters compute the Wronskian of the following equation.
To compute the Wronskian first calculates the roots of the homogeneous portion.
Therefore one of the complimentary solutions is in the form,
where,
Next compute the Wronskian:
Now take the determinant to finish calculating the Wronskian.
Example Question #2 : Variation Of Parameters
Using Variation of Parameters compute the Wronskian of the following equation.
To compute the Wronskian first calculates the roots of the homogeneous portion.
Therefore one of the complimentary solutions is in the form,
where,
Next compute the Wronskian:
Now take the determinant to finish calculating the Wronskian.
Example Question #1 : Variation Of Parameters
Solve the following non-homogeneous differential equation.
Because the inhomogeneity does not take a form we can exploit with undetermined coefficients, we must use variation of parameters. Thus, first we find the complementary solution. The characteristic equation of 
To do variation of parameters, we will need the Wronskian,
Variation of parameters tells us that the coefficient in front of 
and the second half becomes
Putting these together with the complementary solution, we have a general solution of
Example Question #1 : Variation Of Parameters
Find a general solution to the following ODE
None of the other solutions
We know the solution consists of a homogeneous solution and a particular solution.
The auxiliary equation for the homogeneous solution is
The homogeneous solution is
The particular solution is of the form
It requires variation of parameters to solve
Solving the system gets us
Integrating gets us
So
Our solution is
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