Differential Equations Worksheet
Sections 4.1 – 4.6
1)
A fundamental set of solutions of a homogeneous linear nth-order differential equation
comprises
a) Any set of solutions
b) Any set of n solutions
c) Any set of n linearly independent solutions
d) Any set of n homogeneous linearly independent solutions
2) If functions 𝑦1 (𝑥), 𝑦2 (𝑥),,, 𝑦𝑛 (𝑥) form a fundamental set of solutions of a homogeneous linear
nth-order differential equation on interval I, then the Wronskian W(𝑦1 ,𝑦2 , 𝑦𝑛 )
a) Is zero at every x on I
b) May be zero at finitely many x on I
c) Is not =0 at any x on I
d) Is not =0 at any x on I except possibly at x = 0.
3) True or false: A boundary value problem may have no solution
4) True or false: Every nth-order nonhomogeneous linear differential equation has at least one
solution.
1
5) Verify that the functions 𝑦1 = √𝑥 and 𝑦2 = form a fundamental set of solutions of the
√𝑥
6)
7)
8)
9)
10)
11)
12)
13)
14)
differential equation 4𝑥 2 𝑦 ′′ + 4𝑥𝑦 ′ − 𝑦 = 0 on the interval (0,∞)
If 𝑦1 = 𝑒 −𝑥 is a solution of the differential equation 𝑦 ′′ − 2𝑦 ′ − 3𝑦 = 0 use reduction of order
to find a second solution. (You may use the formula)
If 𝑦1 = 𝑥 −2 is a solution of the differential equation 𝑥 2 𝑦 ′′ − 6𝑦 = 0 use reduction of order to
find a second solution. (You may use the formula)
If 𝑦1 = 𝑥 −2 cos⁡(𝑙𝑛𝑥) is a solution of the differential equation⁡⁡⁡𝑥 2 𝑦 ′′ + 5𝑥𝑦 ′ + 5𝑦 = 0 use
reduction of order to find a second solution. (You may use the formula)
If y=xcosx is a solution of a linear homogeneous differential equation then another solution
must be________
Find the general solution of ⁡⁡⁡𝑦 ′′ − 2𝑦 ′ − 3𝑦 = 0
′
Find the general solution of 𝑦 ′′ + 4𝑦 ′′ + 4𝑦 ′ = 0
A certain linear homogeneous DE with constant coefficients has an auxiliary equation whose
roots are m = 0,0,0, -2, -2, ±3i, ±3i. Write the general solution of the DE.
Undetermined coefficients cannot be used if the input function contains what term?
a) 𝑥 2 𝑒 −3𝑥 cos4x
b) 𝑥 −3 cos4x
c) 𝑒 −3𝑥 sin4x
d) 𝑒 −3𝑥 sin2 4x
Without solving the differential equation, apply undetermined coefficients to determine the
simplest form of a particular solution to the DE y’’ +9y’= cos3x
15) Use undetermined coefficients to obtain the general solution of the given differential equation
a) 𝑦 ′′ + 𝑦 ′ − 2𝑦 = 18𝑒 𝑥 − 30𝑐𝑜𝑠𝑥
b) 𝑦 ′′ − 2𝑦 ′ − 3𝑦 = 16𝑒 −𝑥 + 30𝑠𝑖𝑛3𝑥
c) 𝑦 ′′′ − 4𝑦 ′ = 24𝑥 2 + 16𝑐𝑜𝑠2𝑥
16) What operator annihilates sin2x?
17) What operator annihilates x𝑒 2𝑥 ?
18) Use undetermined coefficients (the annihilator approach) to obtain the general solution of the
given differential equation.
a. 𝑦 ′′ + 𝑦 ′ − 2𝑦 = 18𝑒 𝑥 − 30𝑐𝑜𝑠𝑥
b. 𝑦 ′′ + 𝑦 ′ − 2𝑦 = 18𝑒 𝑥 + 4𝑥
c. 𝑦 ′′′ − 4𝑦 ′ = 24𝑥 2 + 16𝑐𝑜𝑠2𝑥
19) We use variation of parameters to obtain
a) The complementary solution of a linear differential equation
b) A particular solution of a linear differential equation
c) Both a and b
d) Neither a nor b
20) True or False: We may use variation of parameters only to solve linear differential equations
that have constant coefficients
21) True or False: When using variation of parameters, the resulting particular solution will always
be linearly independent of the complementary solution.
22) Use variation of parameters to obtain the general solution of the given differential equation
a) 𝑦 ′′ − 4𝑦 = 6 − 12𝑥 2 + 4𝑒 2𝑥
b) 𝑦 ′′ + 2𝑦 ′ = 6𝑒 −2𝑥
c) 𝑦 ′′ + 𝑦 ′ − 6𝑦 = 50𝑥𝑒 2𝑥