At a Glance - More About Solutions
Differential equations have two kinds of solutions: general and particular.
The general solution to a differential equation is the collection of all solutions to that differential equation. A general solution will usually contain some undetermined constants.
Sample Problem
y = 4x + C is the general solution to the d.e.
y ' = 4.
Sample Problem
y = 3x^{2} + Bx + C is the general solution to the d.e.
y " = 6.
A particular solution to a differential equation is a solution with all the constants filled in.
Sample Problem
The function y = 4x + 2 is a specific solution to the d.e.
y ' = 4.
Initial value problems usually have a particular solution only, because the initial condition(s) force us to pick values for the constant(s) in the general solution.
Example 1
Find a particular solution to the IVP y' = 4 and y(1) = 9. |
Example 2
Find the particular solution to the IVP y' = 6x and y(0) = 3. |
Example 3
Solve the IVP y" = 12x and y'(0) = 5 and y(0) = 4 |
Example 4
Solve the IVP f ^{(3)}(x) = 0, f (0) = 0, f '(0) = 1, f "(0) = 4. |
Exercise 1
Solve the initial value problem.
y' = 4x^{3}, y(2) = 20
Exercise 2
Solve the initial value problem.
y' = -sin x, y(0) = 2
Exercise 3
Solve the initial value problem.
y' = x^{2} + x^{3} + 2, y(1) = 3
Exercise 4
Solve the initial value problem.
y" = 4x, y(0) = 7, y'(0) = 5
Exercise 5
Solve the initial value problem.
y' = -e^{x}, y(0) = -3