How To Solve An Initial Value Problem

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A differential equation is called an ordinary differential equation, abbreviated by ode, if it has ordinary derivatives in it.\[\beginay'' by' cy = g\left( t \right)\label\end\] \[\begin\sin \left( y \right)\frac = \left( \right)\frac \label\end\] \[\begin 10y''' - 4y' 2y = \cos \left( t \right) \label\end\] \[\begin\frac = \frac\label\end\] \[\begin = \label\end\] \[\begin\frac = 1 \frac \label\end\] The order of a differential equation is the largest derivative present in the differential equation.In the differential equations listed above \(\eqref\) is a first order differential equation, \(\eqref\), \(\eqref\), \(\eqref\), \(\eqref\), and \(\eqref\) are second order differential equations, \(\eqref\) is a third order differential equation and \(\eqref\) is a fourth order differential equation.\[\begina = \frac\hspace\hspace\,\,\,\,\,\,a = \frac \label\end\] 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.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.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.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.


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