Initial Value Problem - Examples

Examples

A simple example is to solve and . We are trying to find a formula for that satisfies these two equations.

Start by noting that, so

Now rearrange the equation so that is on the left and on the right

Now integrate both sides (this introduces an unknown constant ).

Eliminate the

Let be a new unknown constant, so

Now we need to find a value for . Use as given at the start and substitute 0 for and 19 for

this gives the final solution of .

Second example

The solution of

can be found to be

Indeed,

\begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\ &= 6t+5. \end{align}

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