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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.

' = 4.

Sample Problem

y = 3x2 + Bx + C is the general solution to the d.e.

" = 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.

' = 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' = 4x3, y(2) = 20


Exercise 2

Solve the initial value problem.

y' = -sin x, y(0) = 2


Exercise 3

Solve the initial value problem.

y' = x2 + x3 + 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' = -ex, y(0) = -3


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