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1. Consider the differential equation
(a) Find the homogeneous solution y h(t).
(b) Use the method of unde termined coefficients to find a
particular solution yp(t) to
the differential equation.
(cont’d->)
(c) Write down the general solution y(t) to the
differential equation.
(d) Given the initial conditions y(0) = 1 and y′(0) = 0,
find the specific solution to
the initial value problem .
2. Consider the homogeneous Cauchy-Euler equation
(a) Find the general solution.
(b) Verify that the two homogeneous solutions are linearly independent .
3. Find a general solution y(x) to the fol lowing differential equation:
4. 
is the solution to the homogeneous
differential equation
For each non-homogeneous differential
equation given below, write
down the corresponding form for the particular solution yp(t)
that would be used in
the method of undetermined coefficients . If the method of undetermined
coefficients
does not apply, then write “VoP” to indicate you would use variation of
parameters.
5. Consider the differential equation
which has two linearly independent homogeneous solutions
(a) Find a particular solution yp(t) to
the differential equation using variation of
parameters.
(b) Write down the general solution.
6. Consider non-constant coefficient, homogeneous differential equation
(a) Verify that y1(t) = et is a
solution.
(b) Find a second linearly independent solution y2(t)
using reduction of order .
(c) Write down the general solution.
Formulas
• Quadratic Formula
If ar^2 + br + c = 0, then
• Wronskian
The Wronskian of two differentiable functions y1(t)
and y2(t) is
given by
• Variation of Parameters
Suppose p(t), q(t), and g(t) are continuous. A particular solution y(t) to the
differential
equation
with linearly independent homogeneous solutions y1(t)
and y2(t), has
the form
where
and
• Euler’ s Formula
If θ is any real number , then
