How To Solve An Initial Value Problem

How To Solve An Initial Value Problem-67
\[\begin F = ma \label \end\] To see that this is in fact a differential equation we need to rewrite it a little.First, remember that we can rewrite the acceleration, \(a\), in one of two ways.

\[\begin F = ma \label \end\] To see that this is in fact a differential equation we need to rewrite it a little.First, remember that we can rewrite the acceleration, \(a\), in one of two ways.\[y'\left( x \right) = - \frac\hspacey''\left( x \right) = \frac\] Plug these as well as the function into the differential equation.

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The first definition that we should cover should be that of differential equation.

A differential equation is any equation which contains derivatives, either ordinary derivatives or partial derivatives.

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.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.As you will see most of the solution techniques for second order differential equations can be easily (and naturally) extended to higher order differential equations and we’ll discuss that idea later on.A differential equation is called an ordinary differential equation, abbreviated by ode, if it has ordinary derivatives in it.\[\begina = \frac\hspace\hspace\,\,\,\,\,\,a = \frac \label\end\] Where \(v\) is the velocity of the object and \(u\) is the position function of the object at any time \(t\).We should also remember at this point that the force, \(F\) may also be a function of time, velocity, and/or position.Also note that neither the function or its derivatives are “inside” another function, for example, \(\sqrt \) or \(\).The coefficients \(\left( t \right),\,\, \ldots \,\,,\left( t \right)\) and \(g\left( t \right)\) can be zero or non-zero functions, constant or non-constant functions, linear or non-linear functions.Why then did we include the condition that \(x 0\)?We did not use this condition anywhere in the work showing that the function would satisfy the differential equation.

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