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:

    “It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
    —G.C. (Georg Christoph)

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

    “In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
    —Michel de Montaigne (1533–1592)