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DSolve

DSolve
solves a differential equation for the function y, with independent variable x.
DSolve
solves a list of differential equations.
DSolve
solves a partial differential equation.
  • DSolve gives solutions for 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.
  • For partial differential equations, DSolve typically generates arbitrary functions C[n][...]. »
  • Boundary conditions can be specified by giving equations such as .
  • 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 :
Substitute the solution into an expression:
Solve a differential equation:
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Include a boundary condition:
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Get a "pure function" solution for :
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Substitute the solution into an expression:
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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:
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:
Use differently named constants:
Use subscripted constants:
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^(th) symmetric power of the Legendre differential operator:
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