Solve Differential Equation

通过使用分析地解决微分方程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 usingsyms创建符号函数y(t)。

syms y(t)

Define the equation using==并表示使用的差异化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，因为没有指定条件。用初始条件解决方程y(0) == 2。这dsolvefunction 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

解决具有三个初始条件的三阶微分方程。

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

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 +（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/(3x) + C3x^(1/2)
这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) = C1airy(0,x) + C2airy(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)

这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)