Page 45 - Advanced engineering mathematics

1.3 Exact Equations 25

The “constant of integration” is h(x), because x is held fixed in a partial derivative with respect
to y. Now we know ϕ(x, y) to within h(x). Next we need
∂ϕ

= cos(x) − 2xy =−2xy + h (x).
∂x
This requires that
h (x) = cos(x),

so h(x) = sin(x). A potential function is
y
2
ϕ(x, y) = e − x y + sin(x).
The general solution is implicitly defined by
2
y
e − x y + sin(x) = c.
For the initial condition, choose c so that y(1) = 4. We need
4
e − 4 + sin(1) = c.
The solution of the initial value problem is implicitly defined by
e − x y + sin(x) = e − 4 + sin(1).
4
2
y
SECTION 1.3 PROBLEMS

y
In each of Problems 1 through 5, test the differential equa- 10. 1 + e y/x − e y/x + e y/x y = 0; y(1) =−5

x
tion for exactness. If it is exact (on some region of the
11. x cos(2y − x) − sin(2y − x) − 2x cos(2y − x)y = 0;
plane), find a potential function and the general solution
y(π/12) = π/8
(perhaps implicitly defined). If it is not exact anywhere, do
y
y

not attempt a solution. 12. e + (xe − 1)y = 0; y(5) = 0

13. Let ϕ be a potential function for M + Ny = 0. Show
2
xy
xy
1. 2y + ye + (4xy + xe + 2y)y = 0

that ϕ + c is also a potential function for any con-
2
2
2. 4xy + 2x + (2x + 3y )y = 0 stant c. How does the general solution obtained using

2
2
2

3. 4xy + 2x y + (2x + 3y )y = 0 ϕ differ from that obtained using ϕ + c?

4. 2cos(x + y) − 2x sin(x + y) − 2x sin(x + y)y = 0 If M + Ny = 0 is not exact, it might be
possible to find a nonzero function μ(x, y) such
2

5. 1/x + y + (3y + x)y = 0
that μM + μNy = 0 is exact. The benefit to this

In each of Problems 6 and 7, determine α so that the is that M + Ny = 0and μ(M + Ny ) = 0have
equation is exact. Obtain the general solution of the exact the same solutions if μ(x, y) = 0forany x and
equation. y, and the latter equation is exact (hence is solv-
able if we can find a potential function). Such a
α
2
2
6. 3x + xy − x y α−1 y = 0 function μ(x, y) is called an integrating factor for

2
2
3

7. 2xy − 3y − (3x + αx y − 2αy)y = 0 M + Ny = 0.
14. (a) Show that y −xy =0 is not exact on any rectangle

In each of Problems 8 through 11, determine if the differ- in the plane.
ential equation is exact in some rectangle containing the
(b) Show that μ(x, y) = x −2 is an integrating fac-
point where the initial condition is given. If it is exact,
tor on any rectangle over which x = 0. Use this
solve the initial value problem. If not, do not attempt a
to find the general solution of the differential
solution.
equation.
2
2
2
2
2
8. 2y − y sec (xy ) + (2x −2xy sec (xy ))y = 0; (c) Show that ν(x, y)= y −2 is also an integrating fac-
y(1) = 2 tor on any rectangle where y = 0, and use this to
4
3
9. 3y − 1 + 12xy y = 0; y(1) = 2 solve the differential equation.

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October 14, 2010 14:9 THM/NEIL Page-25 27410_01_ch01_p01-42