Download Review Problems for Test 1

MATH 3160 F15 Review 1 Problems
1
Euler’s Equation
1. Solve x2 y 00 + 5xy 0 + 3y = 0; Ans. y = C1 x−5 C2 x−1 .
2. Solve x2 y 00 + 3xy 0 + y = 0; Ans. y = C1 x−1 + C2 x−1 ln(x).
3. Solve (x − 3)2 y 00 + 5(x − 3)y 0 + 4y = 0; Ans. y = C1 (x − 3)−2 + C2 (x − 3)−2 ln(x − 3).
4. Classify the singular pts of the given equation x2 (2 − x)2 y 00 + 4xy 0 + 5y = 0; Ans. x = 2
regular, x = 0 regular singular.
Method of Frobenius
1. Find two series solutions to the equation 2xy 00 + y 0 − 4y = 0.
P
1 P∞
8n n
8n
n
2
Ans. y1 = ∞
n=0 2n! x , y2 = x
n=0 (2n+1)! x .
2. Find two series solutions to 2x2 y 00 − 3xy 0 + (2 + 2x)y = 0;
P
P∞
1
2n xn
2n xn
n
n+1
2
Ans. y1 = x2 (1+ ∞
).
n=1 (−1) (n!(5·7···(2n+3))) ), y2 = x (1+2x+
n=2 (−1)
(n!(1·3···(2n−3)))
3. Show that x = 0 is a regular singular point to the Laguerre equation xy 00 + (1 − x)y 0 +
py = 0. Show that r = 0 is a double root for the indicial equation and compute the
recurrence formula for a series solution to this differential equation about x = 0.
(r + n)2 an − (r + n − 1 − p)an−1 = 0, n ≥ 1.
Ans.
Bessel’s Equation
1
3
1. Airy’s equation is y 00 + xy = 0. Show that if we set u = x− 2 y and t = 23 x 2 , then Airy’s
equation becomes t2 u00 + tu0 + (t2 − 19 )u = 0. Use this to find the general solution of
Airy’s equation.
3
1
3
Ans. y = x 2 (C1 J 1 ( 2x32 + C2 J− 1 ( 2x32 )).
3
3
2. Using the substitution y = u(µx), find the general solution to the equation x2 y 00 +xy 0 +
(µ2 x2 − p2 )y = 0.
Ans.
y = C1 Jp (µx) + C2 Yp (µx).
3. Find the general solution to x2 y 00 + (x2 − 2)y = 0.
1
Ans. y = x 2 (C1 J 3 (x) + C2 Y 3 (x)).
2
2
Laplace Transforms
−1
1. Find L
n
2−e−2s
s2 +2s+2
o
. Ans. 2e−t sin(t) − U (t − 2)e−(t−2) sin(t − 2).
MATH 3160 F15 Review 1 Problems
2
2. Let [t] denote the greatest integer function, being the least integer greater than or
1
equal to t. eg. [3.3] = 4, [5] = 5. Let f (t) = [t], t ≥ 0. Show that L {f } = s(1−e
−s ) .
Hint: f (t) = t − h(t) where h(t) is a certain periodic function.
n
o
s
−1
3. Use convolution to find L
.
(s−1)(s2 +1)
− 12 cos(t) + 12 sin(t).
nR
o
t
1
4. Find L 0 uet−u du . Ans. s2 (s−1)
.
Ans.
1 t
e
2
5. Solve the following initial-value problem:
y 00 + 4y 0 + 13y = δ(t − π) + δ(t − 3π), y(0) = 1, y 0 (0) = 0.
Ans. y = e−2t cos(3t)+ 32 e−2t sin(3t)+ 13 e−2(t−π) sin(3(t−π))U (t−π)+ 13 e−2(t−3π) sin(3(t−
3π))U (t − 3π).
6. Solve the initial-value problem
y 00 − 6y 0 + 13y = 0, y(0) = 0, y 0 (0) = −3.
Ans.
y = − 32 e3t sin(2t).
Systems of Linear Differential Equations
1. Solve the following system:
dy
d2 x
+ 3 + 3y = 0
2
dt
dt
d2 x
+ 3y = te−t
dt2
where x(0) = 0, x0 (0) = 2, y(0) = 0.
Ans.
x = 12 t2 + t + 1 − e−t , y = − 31 + 13 e−t + 13 te−t .