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dsolve

solve ordinary differential equations (ODEs) Calling Sequence dsolve(ODE) dsolve(ODE, y(x), options) dsolve({ODE, ICs}, y(x), options) Parameters

 ODE - ordinary differential equation, or a set or list of ODEs y(x) - any indeterminate function of one variable, or a set or list of them, representing the unknowns of the ODE problem ICs - initial conditions of the form y(a)=b, D(y)(c)=d, ..., where {a, b, c, d} are constants with respect to the independent variable options - (optional) depends on the type of ODE problem and method used, for example, series or method=laplace. (See the Examples section.) Description

 • As a general ODE solver, dsolve handles different types of ODE problems. These include the following.
 - Computing closed form solutions for a single ODE (see dsolve/ODE) or a system of ODEs, possibly including anti-commutative variables (see dsolve/system).
 - Solving ODEs or a system of them with given initial conditions (boundary value problems). See dsolve/ICs.
 - Computing formal power series solutions for a linear ODE with polynomial coefficients. See dsolve/formal_series.
 - Computing formal solution for a linear ODE with polynomial coefficients. See dsolve/formal_solution.
 - Computing solutions using integral transforms (Laplace and Fourier). See dsolve/integral_transform.
 - Computing numerical (see dsolve/numeric) or series solutions (see dsolve/series) for ODEs or systems of ODEs.
 • The ODE Analyzer Assistant is a point-and-click interface to the ODE solver routines. Using the assistant, you can compute numeric and exact solutions and plot the solutions. For more information, see dsolve[interactive] and worksheet/interactive/dsolve.
 • To define a derivative, use the diff command or one of the notations explained in Derivative Notation. Examples Solving an ODE

 Define a simple ODE.
 > $\mathrm{ode}≔\mathrm{diff}\left(y\left(x\right),x,x\right)=2y\left(x\right)+1$
 ${\mathrm{ode}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{2}{}{y}{}\left({x}\right){+}{1}$ (1)
 Solve the ODE, ode.
 > $\mathrm{dsolve}\left(\mathrm{ode}\right)$
 ${y}{}\left({x}\right){=}{{ⅇ}}^{\sqrt{{2}}{}{x}}{}{\mathrm{_C2}}{+}{{ⅇ}}^{{-}\sqrt{{2}}{}{x}}{}{\mathrm{_C1}}{-}\frac{{1}}{{2}}$ (2)
 Define initial conditions.
 > $\mathrm{ics}≔y\left(0\right)=1,\mathrm{D}\left(y\right)\left(0\right)=0$
 ${\mathrm{ics}}{≔}{y}{}\left({0}\right){=}{1}{,}{\mathrm{D}}{}\left({y}\right){}\left({0}\right){=}{0}$ (3)
 Solve ode subject to the initial conditions ics.
 > $\mathrm{dsolve}\left(\left\{\mathrm{ics},\mathrm{ode}\right\}\right)$
 ${y}{}\left({x}\right){=}\frac{{3}{}{{ⅇ}}^{\sqrt{{2}}{}{x}}}{{4}}{+}\frac{{3}{}{{ⅇ}}^{{-}\sqrt{{2}}{}{x}}}{{4}}{-}\frac{{1}}{{2}}$ (4) Laplace Transform Method

 Compute the solution using the Laplace transform method.
 > $\mathrm{sol}≔\mathrm{dsolve}\left(\left\{\mathrm{ics},\mathrm{ode}\right\},y\left(x\right),\mathrm{method}=\mathrm{laplace}\right)$
 ${\mathrm{sol}}{≔}{y}{}\left({x}\right){=}{-}\frac{{1}}{{2}}{+}\frac{{3}{}{\mathrm{cosh}}{}\left(\sqrt{{2}}{}{x}\right)}{{2}}$ (5)
 Test whether the ODE solution satisfies the ODE and the initial conditions (see odetest).
 > $\mathrm{odetest}\left(\mathrm{sol},\left[\mathrm{ode},\mathrm{ics}\right]\right)$
 $\left[{0}{,}{0}{,}{0}\right]$ (6) Computing a Series Solution

 Find a series solution for the same problem.
 > $\mathrm{series_sol}≔\mathrm{dsolve}\left(\left\{\mathrm{ics},\mathrm{ode}\right\},y\left(x\right),\mathrm{series}\right)$
 ${\mathrm{series_sol}}{≔}{y}{}\left({x}\right){=}{1}{+}\frac{{3}}{{2}}{}{{x}}^{{2}}{+}\frac{{1}}{{4}}{}{{x}}^{{4}}{+}{O}{}\left({{x}}^{{6}}\right)$ (7)
 > $\mathrm{odetest}\left(\mathrm{series_sol},\left[\mathrm{ode},\mathrm{ics}\right],\mathrm{series}\right)$
 $\left[{0}{,}{0}{,}{0}\right]$ (8) Solving an ODE System

 Define a system of ODEs.
 > $\mathrm{sys_ode}≔\mathrm{diff}\left(y\left(t\right),t\right)=x\left(t\right),\mathrm{diff}\left(x\left(t\right),t\right)=-x\left(t\right)$
 ${\mathrm{sys_ode}}{≔}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right){=}{x}{}\left({t}\right){,}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){=}{-}{x}{}\left({t}\right)$ (9)
 If the unknowns are not specified, all differentiated indeterminate functions in the system are treated as the unknowns of the problem.
 > $\mathrm{dsolve}\left(\left[\mathrm{sys_ode}\right]\right)$
 $\left\{{x}{}\left({t}\right){=}{\mathrm{_C2}}{}{{ⅇ}}^{{-}{t}}{,}{y}{}\left({t}\right){=}{-}{\mathrm{_C2}}{}{{ⅇ}}^{{-}{t}}{+}{\mathrm{_C1}}\right\}$ (10)
 Define initial conditions.
 > $\mathrm{ics}≔x\left(0\right)=1,y\left(1\right)=0$
 ${\mathrm{ics}}{≔}{x}{}\left({0}\right){=}{1}{,}{y}{}\left({1}\right){=}{0}$ (11)
 Solve the system of ODEs subject to the initial conditions ics.
 > $\mathrm{dsolve}\left(\left[\mathrm{sys_ode},\mathrm{ics}\right]\right)$
 $\left\{{x}{}\left({t}\right){=}{{ⅇ}}^{{-}{t}}{,}{y}{}\left({t}\right){=}{-}{{ⅇ}}^{{-}{t}}{+}{{ⅇ}}^{{-1}}\right\}$ (12) Details

 • For detailed information on the dsolve command, see dsolve/details.