Solve Differential Equation

通过使用dsolvefunction, with or without initial conditions. To solve a system of differential equations, seeSolve a System of Differential Equations.

First-Order Linear ODE

Solve this differential equation.

$\frac{d y}{d t} = t y .$

First, representyby usingsymsto create the symbolic functiony(t).

syms y(t)

Define the equation using==并代表使用difffunction.

ode = diff(y,t) == t*y

ode(t) = diff(y(t), t) == t*y(t)

Solve the equation usingdsolve.

ySol(t) = dsolve(ode)

ySol(t) = C1*exp(t^2/2)

解微分方程的条件

In the previous solution, the constantC1出现是因为没有指定条件。用初始条件求解方程y(0) == 2. Thedsolvefunction finds a value ofC1满足条件。

cond = y(0) == 2; ySol(t) = dsolve(ode,cond)

ySol(t) = 2*exp(t^2/2)

Ifdsolvecannot solve your equation, then try solving the equation numerically. SeeSolve a Second-Order Differential Equation Numerically.

Nonlinear Differential Equation with Initial Condition

Solve this nonlinear differential equation with an initial condition. The equation has multiple solutions.

$\begin{array}{l} {(\frac{d y}{d t} + y)}^{2} = 1, \\ y (0) = 0。 \end{array}$

syms y(t) ode = (diff(y,t)+y)^2 == 1; cond = y(0) == 0; ySol(t) = dsolve(ode,cond)

ySol(t) = exp(-t) - 1 1 - exp(-t)

Second-Order ODE with Initial Conditions

Solve this second-order differential equation with two initial conditions.

$\begin{array}{l} \frac{d^{2} y}{d x^{2}} = \cos (2 x) - y, \\ y (0) = 1, \\ y' (0) = 0。 \end{array}$

Define the equation and conditions. The second initial condition involves the first derivative ofy. Represent the derivative by creating the symbolic functionDy = diff(y)和then define the condition usingDy(0)==0.

syms y(x) Dy = diff(y); ode = diff(y,x,2) == cos(2*x)-y; cond1 = y(0) == 1; cond2 = Dy(0) == 0;

Solveodefory. Simplify the solution using the简化function.

conds = [cond1 cond2]; ySol(x) = dsolve(ode,conds); ySol = simplify(ySol)

ySol(x) = 1 - (8*sin(x/2)^4)/3

Third-Order ODE with Initial Conditions

Solve this third-order differential equation with three initial conditions.

$\begin{array}{l} \frac{d^{3} u}{d x^{3}} = u, \\ u (0) = 1, \\ u^{'} (0) = - 1, \\ {u^{'}}^{'} (0) = π . \end{array}$

Because the initial conditions contain the first- and second-order derivatives, create two symbolic functions,Du = diff(u,x)和d2u = diff（u，x，2）, specify the initial conditions.

syms u(x) Du = diff(u,x); D2u = diff(u,x,2);

Create the equation and initial conditions, and solve it.

ode = diff(u,x,3) == u; cond1 = u(0) == 1; cond2 = Du(0) == -1; cond3 = D2u(0) == pi; conds = [cond1 cond2 cond3]; uSol(x) = dsolve(ode,conds)

uSol(x) = (pi*exp(x))/3 - exp(-x/2)*cos((3^(1/2)*x)/2)*(pi/3 - 1) -... (3^(1/2)*exp(-x/2)*sin((3^(1/2)*x)/2)*(pi + 1))/3

More ODE Examples

This table shows examples of differential equations and their Symbolic Math Toolbox™ syntax. The last example is the Airy differential equation, whose solution is called the Airy function.

Differential Equation	MATLAB^®Commands
$\begin{array}{l} \frac{d y}{d t} + 4 y (t) = e^{- t}, \\ y (0) = 1. \end{array}$	syms y(t) ode = diff(y)+4y == exp(-t); cond = y(0) == 1; ySol(t) = dsolve(ode,cond) ySol(t) = exp(-t)/3 + (2exp(-4*t))/3
$2 x^{2} \frac{d^{2} y}{d x^{2}} + 3 x \frac{d y}{d x} - y = 0。$	syms y（x）ode = 2x^2diff（y，x，2）+3xdiff（y，x）-y == 0;ysol（x）= dsolve（ode） ySol(x) = C2/(3x) + C3x^(1/2)
The Airy equation. $\frac{d^{2} y}{d x^{2}} = x y (x) .$	syms y（x）ode = diff（y，x，2）== xy;ysol（x）= dsolve（ode） ySol(x) = C1airy(0,x) + C2*airy(2,x)

Differential Equation

MATLAB^®Commands

$\begin{array}{l} \frac{d y}{d t} + 4 y (t) = e^{- t}, \\ y (0) = 1. \end{array}$

syms y(t) ode = diff(y)+4*y == exp(-t); cond = y(0) == 1; ySol(t) = dsolve(ode,cond)

ySol(t) = exp(-t)/3 + (2*exp(-4*t))/3

$2 x^{2} \frac{d^{2} y}{d x^{2}} + 3 x \frac{d y}{d x} - y = 0。$

syms y（x）ode = 2*x^2*diff（y，x，2）+3*x*diff（y，x）-y == 0;ysol（x）= dsolve（ode）

ySol(x) = C2/(3*x) + C3*x^(1/2)

The Airy equation.

$\frac{d^{2} y}{d x^{2}} = x y (x) .$

syms y（x）ode = diff（y，x，2）== x*y;ysol（x）= dsolve（ode）

ySol(x) = C1*airy(0,x) + C2*airy(2,x)