Tutorial 2: Worksheet
Goals.
(1) Any questions?
(2) Practice solving 2nd-order linear ODEs with constant coefficients
• Distinct real roots
• Repeated real roots
• Particular solutions
(3) Simpifying non-homogenous equations using exponential substitutions
Distinct Real Roots.
(1) Find a particular solution of y 00 − 7y 0 + 12y = 4e2x . Hint: Try yp = something that
looks like the right hand side.
(2) Find the general solution. Hint: What is the characteristic polynomial?
(3) Solve the IVP y 00 − 7y 0 + 12y = 4e2x , y(0) = α, y 0 (0) = β. Hint: Solve for c1 and
c2 .
(4) Find a particular solution of y 00 − 7y 0 + 12y = 5e4x by looking for a solution of the
form y = ue4x where u is a function to be determined.
(5) What is the general solution in this case?
Repeated Real Roots.
(1) Find the general solution of y 00 + 6y 0 + 9y = 0.
(2) Explain why yp1 = Ae−3x and yp2 = Bxe−3x are not particular solutions of y 00 +
6y 0 + 9y = 10e−3x .
(3) Find a particular solution of the above equation. Hint: try y = ue−3x .
(4) Find the general solution to y 00 + 6y 0 + 9y = 10e−3x .
(5) What is limx→∞ y(x)?
Conclusions. Consider ay 00 + by 0 + cy = keαx , with k 6= 0, a, b, c, ∈ R.
.
(1) If eαx is not a solution of ay 00 + by 0 + cy = 0, then yp =
(2) If eαx is a solution but xeαx is not a solution of ay 00 + by 0 + cy = 0, then yp =
(3) If eαx and xeαx are solutions of ay 00 + by 0 + cy = 0, then yp =
.
.
Harder. Consider ay 00 + by 0 + cy = keαx G(x), where G is a polynomial of degree greater
than 0.
(1) Show that au00 + p0 (α)u0 + p(α)u = G(x) where p(r) = ar2 + br + c and u = u(x)
is an unknown function.
(2) Find a particular solution to y 00 − 4y = e−x (7 − 4x + 5x2 ).
Next tutorial: Complex numbers, characteristic equations with complex roots, sines and
cosines. Why do we do all of this? Answer: mass-spring oscillator.
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