How To Solve An Initial Value Problem

How To Solve An Initial Value Problem-23
There is one differential equation that everybody probably knows, that is Newton’s Second Law of Motion.If an object of mass \(m\) is moving with acceleration \(a\) and being acted on with force \(F\) then Newton’s Second Law tells us.Also note that neither the function or its derivatives are “inside” another function, for example, \(\sqrt \) or \(\).

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A linear differential equation is any differential equation that can be written in the following form.

\[\begin \left( t \right)\left( t \right) \left( t \right)\left( t \right) \cdots \left( t \right)y'\left( t \right) \left( t \right)y\left( t \right) = g\left( t \right) \label\end\] The important thing to note about linear differential equations is that there are no products of the function, \(y\left( t \right)\), and its derivatives and neither the function or its derivatives occur to any power other than the first power.

Note that the order does not depend on whether or not you’ve got ordinary or partial derivatives in the differential equation.

We will be looking almost exclusively at first and second order differential equations in these notes.

In \(\eqref\) - \(\eqref\) above only \(\eqref\) is non-linear, the other two are linear differential equations.

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We can’t classify \(\eqref\) and \(\eqref\) since we do not know what form the function \(F\) has.

Eriksson, K.; Estep, D.; Hansbo, P.; and Johnson, C.

"Initial Value Problems." Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.

Only the function,\(y\left( t \right)\), and its derivatives are used in determining if a differential equation is linear.

If a differential equation cannot be written in the form, \(\eqref\) then it is called a non-linear differential equation.


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