Introduction to Differential Equations Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy
Verify that y=sin⁡x+2cos⁡xy = \sin x + 2\cos x satisfies y′′+y=0y'' + y = 0.

Solution

  1. 1
    y′′=−sin⁡x−2cos⁡xy'' = -\sin x - 2\cos x.
  2. 2
    y′′+y=(−sin⁡x−2cos⁡x)+(sin⁡x+2cos⁡x)=0y''+y = (-\sin x-2\cos x)+(\sin x+2\cos x) = 0. ✓

Answer

Verified.
Differentiate twice and substitute; the terms cancel.

About Introduction to Differential Equations

An equation that contains an unknown function and one or more of its derivatives. Solving a DE means finding the function(s) that satisfy the equation.

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More Introduction to Differential Equations Examples

Example 1 easy

Verify [formula] solves [formula], and find the particular solution with [formula].

Example 2 medium

Find the general and particular ([formula]) solution to [formula].

Example 4 medium

Find the particular solution to [formula] with [formula].

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