**Unformatted text preview: **y = xe x + Ce x is a solution of
dx
dy
the differential equation
= e x + y . (See Example 2.2.)
dx 1. By computing 2. Verify that for any constant C the expression y = x 2 − 2 x + C is a solution of the differential
dy x − 1
=
.
equation
dx
y
3. a) Verify that for any constant C the expression y = x − 1 + Ce − x is a solution of the
dy
differential equation
= x− y.
dx
b) Find the solution of the differential equation in 3a) that satisfies y (0) = 2 .
4. a) Verify that for any constant C the expression y =
equation dy 1 + xy
=
.
dx 1 + x Ce x − 1
is a solution of the differential
1+ x b) Find the solution of the differential equation in 4a)that satisfies y (1) = 2 .
5. a) Verify that for any constant C ≥ 0 the expression y = − x + 2 x 2 + C is a solution of the
dy x − y
differential equation
=
.
dx x + y
b) Find the solution of the differential equation in 5a) that satisfies y (1) = 1 .
6. Which, if any, of the differential equations in exercises 1 - 5 are separable?
7. Which, if any, of the following differential equations are separable?
27 2 Differential Equations
a) dy
xy
=
dx 1 + x b) dy
xy
=
dx y + x c) dy
= (3 y + 1) 2
dx d) dy 1 + y 2
=
dx 1 + x 2 Use separation of variables to solve each of the differential equations in exercises 8 - 16 below:
8. dy
= xy , y (0) = 2 .
dx 9. dy
= 2 y 2 x , y (0) = 2
dx 10. dy
=
dt 11. dy
t
= 2 y , y (0) = 1 .
dt e 12. dy
= 3 y + 1 , y (1) = 2 .
dt 13. y′ = y ( x + 1) , y (0) = 2 . 14. dy
2y
=
, y (0) = 3 .
dx x + 1 15. y′ = 16. dy 1 + 2 y
=
dx 1 + x t
, y (1) = 3
y +1 2(t + 1)
, y (0) = 2 .
y 17. After 5 years, which of the following accounts will have a larger principal? After 10 years?
Justify your answers.
a) An account earning 5% annual interest compounded continuously and starting out with
$1000.
b) An account earning 4% annual interest compounded continuously and starting out with
$1100. 18. a) Suppose you want to put away a certain amount of money so that in 10 years the principal
will be $10,000. If you can invest at 6% annual interest compounded conti...

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