The function dsolve(d_1,..., d_n, i_1,..., i_m, v), attempts to find a solution for the system of differential equations d_1,..., d_n with initial conditions i_1,..., i_m. The solution is expressed in terms of the independent variable v.  If the optional argument v is omitted, then the independent variable is t.

For a given system of equations and set of initial conditions, dsolve will attempt to find an explicit solution.  If no suitable explicit solution can be found, then dsolve will display a warning and return an implicit solution.

The format of the input equations differs from the standard version of dsolve in two ways.  If y is a function of a single variable, then:

. The independent variable should not be listed explicitly. That is, instances of y should appear as y and not y(x).

. The nth derivative of y should appear as Dny. For example, D2y for the second derivative of y. For n = 1, Dy is also accepted.

Note: since the independent variable is implied, only variables whose derivatives appear in the input equations are considered to be functions.

The format of the returned solutions is determined by the number of dependent variables defined in the equations, and by the number of output arguments.

When the complete solution is explicit, the solutions for a given function are returned as a vector of expressions when there are multiple solutions, or as a single expression when there is only one solution. If the system contains multiple functions, then:

. If the number of output arguments is equal to the number of functions, dsolve will return an expression sequence of solution vectors (or expressions), sorted in lexicographical order by function name.

. If there are zero or one output arguments, dsolve will return a MTM[struct], R, where R:-f is the solution vector (or expression) for the function f.

Note: when solutions for the individual functions in a system are returned as vectors, the set of ith elements of the vectors compose a single solution of the system.

When the complete solution is implicit, if the system contains only one function:

. If there is only one implicit solution, a single equation is returned.

. If there are multiple implicit solutions, then the equations are returned as a vector.

If the system contains multiple functions, then:

. If the number of output arguments is equal to the number of functions, dsolve will return an expression sequence of vectors containing the implicit equations for each function. In this case, the set of ith elements of the vectors compose a single implicit solution of the system.

. If there are zero or one output arguments, dsolve will return a matrix of equations. In this case, each row is a single implicit solution of the system.

$\mathrm{with}\left(\mathrm{MTM}\right)\:$

$\mathrm{dsolve}\left(\mathrm{Dx}=-ax\right)$

$\mathrm{c\_\_1}{\ⅇ}^{-at}$

$\mathrm{dsolve}\left({\mathrm{Dy}}^2+y^2=1\right)$

$\left[\begin{array}{c}\mathrm{-1}\\ 1\\ -\sin \left(-t+\mathrm{c\_\_1}\right)\\ \sin \left(-t+\mathrm{c\_\_1}\right)\end{array}\right]$

$S≔\mathrm{dsolve}\left({\mathrm{Dy}}^2+y^2=1\,\mathrm{Dx}=0\,x=1\right)$

$S≔\mathrm{x:\; [\; 4x1\; Vector[column]\; ]\; y:\; [\; 4x1\; Vector[column]\; ]}$

$X,Y≔\mathrm{dsolve}\left(\mathrm{Dy}=y^2\left(1-y\right)y\,\mathrm{Dx}=4\right)$

$X,Y≔x\left(t\right)=4t+\mathrm{c\_\_2},t+\frac{1}{2{y\left(t\right)}^2}+\frac{1}{y\left(t\right)}-\ln \left(y\left(t\right)\right)+\ln \left(y\left(t\right)-1\right)+\mathrm{c\_\_1}=0$

$\mathrm{dsolve}\left(\mathrm{Dy}=y^2\left(1-y\right)y\,\mathrm{Dx}=4\right)$

$\left[\begin{array}{cc}x\left(t\right)=4t+\mathrm{c\_\_2}& t+\frac{1}{2{y\left(t\right)}^2}+\frac{1}{y\left(t\right)}-\ln \left(y\left(t\right)\right)+\ln \left(y\left(t\right)-1\right)+\mathrm{c\_\_1}=0\end{array}\right]$