Differential Equations

The function DSolve is used to solve differential equations. Note the following examples

In[38]:=	DSolve[y'[x] == a y[x], y[x], x]
Out[38]=	$${\{\{y[x]\mathrel{-\negthinspace\negthinspace>}E^{ax} C[1]\}\} }$$
In[39]:=	DSolve[{y'[x] == a y[x], y[0] == 1}, y[x], x]
Out[39]=	$${\{\{y[x]\mathrel{-\negthinspace\negthinspace>}E^{ax}\}\} }$$

where we can see several important features. The notation y'[x] has the obvious meaning of 1st derivative of the function y[x] with respect to x. The 2nd and 3rd parameters of the DSolve function are the unknown function and the variable of integration respectively. Note also the appearance of the arbitrary constant $C[1]$ in the 1st example and the inclusion of a boundary condition y[0] == 1 in the 2nd.

Mathematica can handle more general systems of differential equations, but solutions are usually only found if the equations are linear, unless the user is able to fill in some intermediate steps.

There are also functions NSolve and NDSolve which provide numerical solutions when the analytical solutions are not available from Solve or DSolve.