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# 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.
• 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 , which yields constants of integration C[1], C[2], ... .  »
• 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 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.  »
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:
 Out[1]=
Include a boundary condition:
 Out[2]=

Get a "pure function" solution for y:
 Out[1]=
Substitute the solution into an expression:
 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:
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:
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:
Results may contain symbolic integrals:
Inverse functions may be required to find the solution:
Generate a Cornu spiral:
Solve the 6 symmetric power of the Legendre differential operator: