Download Solved Problems on Elementary Differential Equations - Assignment 5 | Math 3 and more Assignments Mathematics in PDF only on Docsity!
MATH 3D Homework 5 Name: Solutions
Feb. 10, 2006
hw5.pdf Problems
- Find the general solution of
1 +
t y
− sin(y)
dy dt
Solution: (10 points) This is a first-order, nonlinear problem, and it doesn’t look separable, so we look to the method of exact equations. However, if M = 1 and N =
t y
− sin(y),
then My 6 = Nt, and the equation is not exact. So, we multiply by μ and see if we can make it exact: μ + μ
t y
− sin y
y′^ = 0.
Then we require that
μy =
μ
t y
− sin y
t = μt
t y
− sin y
y
μ.
Suppose that μ = μ(y), and so μt = 0 and μy = μ′. Then
μ′^ −
y
μ = 0 =⇒ μ = e
∫ (^1) y dy^ = eln|y|^ = |y|.
So, the modified equation is
|y| +
t |y| y
− |y| sin y
y′^ = 0.
Suppose first that y > 0. Then the ODE is y + (t − y sin y) y′^ = 0.
If M = y and N = t − y sin y, then we require that ϕt = M = y ϕy = N = t − y sin y.
By the first equation,
ϕ =
ϕt dt = ty + f (y) =⇒ ϕy = t + f ′(y) = t − y sin y.
So, f ′(y) = −y sin y =⇒ f (y) = y cos y − sin y + c So, ϕ(t, y) = ty + y cos y − sin y + c = 0 gives the general solution.
Document URL: http://math.uci.edu/~pmacklin/Math3Dwinter2006.html Date: February 22, 2006 Page 1 of 2
Section 2.1 Problems
- Let y 1 and y 2 be solutions of Bessel’s equation
t^2 y′′^ + ty′^ + (t^2 − n^2 )y = 0
on the interval 0 < t < ∞, with y 1 (1) = 1, y′ 1 (1) = 0, y 2 (1) = 0, and y′ 2 (1) = 1. Compute W y 1 , y 2 .
Solution: (10 points) Recall that if y 1 and y 2 are solutions to
y′′^ + py′^ + qy = 0,
on (a, b), then the Wronskian W (t) = W y 1 , y 2 solves W ′^ + pW = 0 on (a, b). Our differential equation is not in the proper form; we have to first divide by t^2 :
y′′^ +
t
y +
t^2 n^2
y = 0.
Then W solves W ′^ + (^1) t W = 0 on (0, ∞), i.e.,
W (t) = W (1)e−^
∫ (^) t 11 s ds^ = W (1)eln 1−ln^ t^ = W (1)^1 t
Now, we need W (1):
W (1) = det
y 1 (1) y 2 (1) y 1 ′(1) y′ 2 (1)
= y 1 (1)y′ 2 (1) − y 2 (1)y 1 ′(1) = 1.
Thus, W (t) =
t
Completeness Points
(5 points) One point per seriously-attempted, assigned problem.
Document URL: http://math.uci.edu/~pmacklin/Math3Dwinter2006.html Date: February 22, 2006 Page 2 of 2
