Even if you don’t know how to find a solution to a differential equation, you can always check whether a proposed solution works. This is simply a matter of plugging the proposed value of the dependent variable into both sides of the equation to see whether equality is maintained.
For example, here’s a differential equation with a dependent variable y:
You may not have a clue how to begin solving this differential equation, but imagine that an angel lands on your pen and offers you this solution:
y = 4e3x sin x
You can check to see whether this angel really knows math by plugging in this value of y as follows:
Because the left and right sides of the equation are equal, the angel’s solution checks out.
