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}

Read more about this topic:  Initial Value Problem

Famous quotes containing the word examples:

    “There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
    —Bernard Mandeville (1670–1733)

    “No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
    —André Breton (1896–1966)

    “Histories are more full of examples of the fidelity of dogs than of friends.”
    —Alexander Pope (1688–1744)