Built-in Mathematica Symbol

DSolve

DSolve[eqn, y, x]
solves a differential equation for the function y, with independent variable x.

DSolve[{eqn₁, eqn₂, ...}, {y₁, y₂, ...}, x]
solves a list of differential equations.

DSolve[eqn, y, {x₁, x₂, ...}]
solves a partial differential equation.

MORE INFORMATION

DSolve[eqn, y[x], x] gives solutions for y[x] rather than for the function y itself.

Differential equations must be stated in terms of derivatives such as , obtained with D, not total derivatives obtained with Dt.

The list of equations given to DSolve can include algebraic ones that do not involve derivatives.

DSolve generates constants of integration indexed by successive integers. The option GeneratedParameters specifies the function to apply to each index. The default is GeneratedParameters->C, which yields constants of integration C[1], C[2], ... . »

GeneratedParameters->(Module[{C}, C]&) guarantees that the constants of integration are unique, even across different invocations of DSolve.

For partial differential equations, DSolve typically generates arbitrary functions C[n][...]. »

Boundary conditions can be specified by giving equations such as y'[0]b.

Solutions given by DSolve sometimes include integrals that cannot be carried out explicitly by Integrate. Dummy variables with local names are used in such integrals.

DSolve sometimes gives implicit solutions in terms of Solve. »

DSolve can solve linear ordinary differential equations of any order with constant coefficients. It can also solve many linear equations up to second order with nonconstant coefficients.

DSolve includes general procedures that handle almost all the nonlinear ordinary differential equations whose solutions are given in standard reference books such as Kamke.

DSolve can find general solutions for linear and weakly nonlinear partial differential equations. Truly nonlinear partial differential equations usually admit no general solutions.

DSolve can handle not only pure differential equations but also differential-algebraic equations. »

EXAMPLES

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Basic Examples (2)

Solve a differential equation:

Include a boundary condition:

Get a "pure function" solution for y:

Substitute the solution into an expression:

Solve a differential equation:

In[1]:=

Out[1]=

Include a boundary condition:

In[2]:=

Out[2]=

Get a "pure function" solution for y:

In[1]:=

Out[1]=

Substitute the solution into an expression:

In[2]:=

Out[2]=

Scope (24)

Exponential equation:

Inhomogeneous first-order equation:

Solve a boundary value problem:

Plot the solution:

Second-order equation with constant coefficients:

Cauchy-Euler equation:

Second-order equation with variable coefficients, solved in terms of elementary functions:

Airy's equation:

Spheroidal equation:

Equations with nonrational coefficients:

Higher-order equations:

Solution in terms of hypergeometric functions:

Fourth-order equation solved in terms of Kelvin functions:

Using a piecewise forcing function:

A differential equation with a piecewise coefficient:

A nonlinear piecewise-defined differential equation:

Differential equations involving generalized functions:

A simple impulse response or Green's function:

Solve a Riccati equation:

Implicit solution for an Abel equation:

Homogeneous equation:

Solution in terms of WeierstrassP:

Solution in terms of hyperbolic functions:

Diagonal linear system:

Inhomogeneous linear system with constant coefficients:

Nonlinear system:

Solve a system of linear differential-algebraic equations:

Solve a boundary value problem:

An index-2 differential-algebraic equation:

General solution for a linear first-order partial differential equation:

The solution with a particular choice of the arbitrary function C[1]:

General solution for a quasilinear first-order partial differential equation:

Complete integral for a nonlinear, first-order Clairaut equation:

Initial value problem for a linear first-order partial differential equation:

Linear second-order partial differential equation with constant coefficients:

Traveling wave solution for the Korteweg-de Vries (KdV) equation:

Generalizations & Extensions (1)

No boundary condition, gives two generated parameters:

One boundary condition:

Two boundary conditions:

Options (1)

Use differently named constants:

Use subscripted constants:

Applications (7)

Solve a logistic (Riccati) equation:

Plot the solution for different initial values:

Solve a linear pendulum equation:

Displacement of a linear, damped pendulum:

Study the phase portrait of a dynamical system:

Find a power series solution when the exact solution is known:

Recover a function from its gradient vector:

Solve a Cauchy problem to generate Stirling numbers:

Properties & Relations (4)

Solutions satisfy the differential equation and boundary conditions:

Differential equation corresponding to Integrate:

Use NDSolve to find a numerical solution:

Compute an impulse response using DSolve:

The same computation using InverseLaplaceTransform:

Possible Issues (2)

Results may contain symbolic integrals:

Inverse functions may be required to find the solution:

Neat Examples (2)

Generate a Cornu spiral:

Solve the 6

symmetric power of the Legendre differential operator: