MA 2051 D '97 Sample Exam 1
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The function u is a solution if it is continuously differentiable on the interval and satisfies
The characteristic equation is r+2 = 0, so the general solution to the homogeneous equation is
The form for is
so we have
Comparing to the right-hand side of our DE, we get the equations
The solution is A=1/2, B=-1/2, C=1/4, and D=1 so
The general solution is . Evaluating at y=-1 gives
so we have
This DE is linear. If we can write it in our standard form
Separation of variables gives
Integrating, and making the usual argument gives the solution
as long as .
To use VOP, we write and substitute into the left-hand side of the DE.
Setting this equal to the right-hand side and solving for u', we get
Multiplying by gives the general solution as
Simply plugging x=0 into our general solution gives y(0) = 0, so there is no way to match the initial condition. What goes wrong is that the coefficient 1/x is undefined at x=0, so we can't expect to get a solution at x=0.
Using the same letters to denote parameters that we used in class, we get
for the energy balance equation. Dividing both sides by cm gives
One way to solve this IVP is to write the DE in standard linear form as
The homogeneous solution can easily be obtained using the characteristic equation as
and the particular solution can be obtained using the method of undetermined coefficients as
Combining the particular and homogeneous solutions and applying the initial condition, we get the following solution.
The solutions of this model decay exponentially to the constant
Note that this is greater than , because of the heat source Q.