1. Find the function f(x) given that f '(x) = 4x + 3 and f(1) = 2
2. A curve passes through the point (2, 5) and its gradient at any point is given by the function
dy/dx = 2x.
What is the equation of the curve?
3. If f ''(x) = 12x2 + 24x + 12 and f '(2) = 98 and f(1) = 7
4. Find the general solution to the differential equation y ' = x + 1
5. Find the general solution to the differential equation dy/dx = x2 + 4x − 3
6. Find the general solution for the differential equation dy/dx = 2x and draw a few example from the family of curves.
7. Find the general solution to each of the following second order differential equations:
a. f ''(x) = -10
b. d2y/dx2 = 0.5x
c. y '' = x2
d. d2y/dx2 = cos x
e. f ''(x) = 0
For each of the differential equations below two particular solutions are suggested.
Test each solution and state which is correct (or both).
| Differential Equation | Suggestion 1 | Suggestion 2 | |
|
8
|
f '(x) = y | y = ex | y = e-x |
|
9
|
dy/dx = 2x | y = e2x | y = x2 |
|
10
|
y '' = y | y = 2ex | y = 3e-x |
|
11
|
d2y/dx2 + dy/dx = 0 | y = sin x | y = e-x |
