符号解用dsolve函数
例 求微分方程$y''+4y'+4y=e^{-2x}$

 dsolve('D2y+4*Dy+4*y=exp(-2*x)')

结果

ans =1/(4*exp(2*x)) + C5/exp(2*t) + (C6*t)/exp(2*t)

例 求解微分方程$x^2y'+xy=y^2,y(1)=1$的特解

dsolve('x^2*Dy+x*y=y^2','y(1)=1')

结果

ans =-(x*(tan(-(x*((- 1 + 2*x*atan((2*i)/x - i)*i)/x^2 + t/x^2)*i)/2) + i)*i)/2

数值解用龙格库塔方法
函数
$[X,Y]=ode23('f',[x_0,x_n],y_0)$
$[X,Y]=ode45('f',[x_0,x_n],y_0)$
f是函数，$y_0$是初值，$[x_0,x_n]$是区间
ode23是二阶三阶龙格库塔方法，ode45是四阶五阶龙格库塔方法
例 求微分方程

$$\left\{ \begin{aligned} \frac{\rm d y}{\rm d x}=-3y+2x \\ y(0)=1 \end{aligned} \right.$$
在[1,3]区间内的数值解。
f.m

function f=f(x,y)
f=-3.*y+2.*x;
end

输入命令

[X,Y]=ode45('f',[1 3],1);
X',Y'

结果

ans =Columns 1 through 81.0000    1.0500    1.1000    1.1500    1.2000    1.2500    1.3000    1.3500Columns 9 through 161.4000    1.4500    1.5000    1.5500    1.6000    1.6500    1.7000    1.7500Columns 17 through 241.8000    1.8500    1.9000    1.9500    2.0000    2.0500    2.1000    2.1500Columns 25 through 322.2000    2.2500    2.3000    2.3500    2.4000    2.4500    2.5000    2.5500Columns 33 through 402.6000    2.6500    2.7000    2.7500    2.8000    2.8500    2.9000    2.9500Column 413.0000ans =Columns 1 through 81.0000    0.9559    0.9226    0.8987    0.8827    0.8735    0.8703    0.8722Columns 9 through 160.8785    0.8885    0.9017    0.9178    0.9363    0.9568    0.9791    1.0030Columns 17 through 241.0282    1.0545    1.0818    1.1099    1.1388    1.1683    1.1983    1.2287Columns 25 through 321.2596    1.2908    1.3224    1.3541    1.3861    1.4183    1.4506    1.4831Columns 33 through 401.5157    1.5484    1.5812    1.6140    1.6470    1.6799    1.7130    1.7460Column 411.7792

例 求微分方程

$$\left\{ \begin{aligned} 5\frac{\rm d y}{\rm d x}+y=1 \\ y(0)=2 \end{aligned} \right.$$
的数值解。
f.m

function f=f(x,y)
f=-y./5+1/5;
end

输入命令

[X,Y]=ode45('f',[1 3],2);
X',Y'

结果

ans =Columns 1 through 81.0000    1.0500    1.1000    1.1500    1.2000    1.2500    1.3000    1.3500Columns 9 through 161.4000    1.4500    1.5000    1.5500    1.6000    1.6500    1.7000    1.7500Columns 17 through 241.8000    1.8500    1.9000    1.9500    2.0000    2.0500    2.1000    2.1500Columns 25 through 322.2000    2.2500    2.3000    2.3500    2.4000    2.4500    2.5000    2.5500Columns 33 through 402.6000    2.6500    2.7000    2.7500    2.8000    2.8500    2.9000    2.9500Column 413.0000ans =Columns 1 through 82.0000    1.9900    1.9802    1.9704    1.9608    1.9512    1.9418    1.9324Columns 9 through 161.9231    1.9139    1.9048    1.8958    1.8869    1.8781    1.8694    1.8607Columns 17 through 241.8521    1.8437    1.8353    1.8270    1.8187    1.8106    1.8025    1.7945Columns 25 through 321.7866    1.7788    1.7711    1.7634    1.7558    1.7483    1.7408    1.7334Columns 33 through 401.7261    1.7189    1.7118    1.7047    1.6977    1.6907    1.6839    1.6771Column 411.6703