Math 54 Summer 2017 Worksheet 25
Differential Equations
1. Which of the following functions are solutions to the differential equation y00 −y= 2 −t2?
(a) f:R→Rdefined by f(t) = t2
(b) g:R→Rdefined by g(t) = tet
(c) h:R→Rdefined by h(t) = t2+et
2. Which of the functions in problem 1 are solutions to the initial value problem
y00 −y= 2 −t2
y(0) = 1
y0(0) = 1
3. Find the general solution of the following differential equations.
(a) y00 −2y0−3y= 0
(b) y00 −5y0= 0
(c) y000 + 5y00 + 4y0= 0
4. Find a differential equation for which e7t+ 4e−3tis a solution.
5. Find the solution to each of the following initial value problems.
(a) y00 +y0= 0, y(0) = 2, y0(0) = 1
(b) y000 + 5y00 + 4y0= 0, y(0) = 8, y0(0) = −9, y00(0) = 33
Definitions and Theorems
Definitions:
•Differential Equation, ODE
•Solution to a differential equation
•Initial value problem (IVP)
•Linear, constant coefficient, homogeneous
ODE
•Auxiliary equation
Most important idea today: Finding the general solution to a linear, constant coefficient,
homogeneous ODE just means finding the kernel of some linear transformation.
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