• [y1, …, yN ] = dsolve(eqns, conds ) eqns:微分方程（组），conds 为初值条件或边界条件

  求解微分方程的通解

  例1

• 求解常微分方程

  syms y(x)%定义符号变量
  dsolve(x^2+y+(x-2*y)*diff(y)==0)

  求解常微分方程的初边值问题

  例2

  ```matlab syms y(x) dy=diff(y); d2y=diff(y,2); y=dsolve(diff(y,3)-diff(y,2)==x,y(1)==8,dy(1)==7,d2y(2)==4); y=simplify(y)

y =

(17x)/2 + 7exp(x - 2) - 7xexp(-1) - x^2/2 - x^3/6 + 1/6

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# 求解常微分方程组
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## 例3
![](https://cdn.nlark.com/yuque/__latex/e3f67e80cd1e91122658fd67e1b92928.svg#card=math&code=%5Cbegin%7Bcases%7D%0Af%5E%7B%27%27%7D%2B3g%3Dsinx%5C%5C%0Ag%5E%7B%27%7D%2Bf%5E%7B%27%7D%3Dcosx%0A%5Cend%7Bcases%7D%0A%5Cquad%0Af%5E%7B%27%7D%282%29%3D0%2Cf%283%29%3D3%2Cg%285%29%3D1&id=YvFK3)
```matlab
syms f(x) g(x)
df=diff(f);
[f1,g1]=dsolve(diff(f,2)+3*g==sin(x),diff(g)+df==cos(x))
f1=simplify(f1);
g1=simplify(g1);
[f2,g2]=dsolve(diff(f,2)+3*g==sin(x),diff(g)+df==cos(x),df(2)==0,f(3)==3,g(5)==1)
f2=simplify(f2);
g2=simplify(g2);

求解线性常微分方程组

一阶齐次线性微分方程组

例4

• 求解初值问题

syms x1(t) x2(t) x3(t)
X=[x1;x2;x3];
A=[2 1 3;0 2 -1;0 0 2];
B=[1;2;1];
[x1,x2,x3]=dsolve(diff(X)==A*X,X(0)==B)

非齐次线性方程组

例5

• 求解初值问题

syms x1(t) x2(t) x3(t)
X=[x1;x2;x3];
A=[1 0 0;2 1 -2;3 2 1];
B=[0;0;exp(t)*cos(2*t)];
X0=[0;1;1];
%[x1,x2,x3]=dsolve(diff(X)==A*X+B,X(0)==X0)
X=dsolve(diff(X)==A*X+B,X(0)==X0)
X=simplify([X.x1;X.x2;X.x3])%化简
pretty(X)%数学显示
>>> X =
                                                   0
  -(exp(t)*(2*sin(2*t) - 2*cos(2*t) + t*sin(2*t)))/2
 (exp(t)*(4*cos(2*t) + 5*sin(2*t) + 2*t*cos(2*t)))/4
/                        0                        \
|                                                 |
|   exp(t) (sin(2 t) 2 - cos(2 t) 2 + t sin(2 t)) |
| - --------------------------------------------- |
|                         2                       |
|                                                 |
| exp(t) (cos(2 t) 4 + sin(2 t) 5 + t cos(2 t) 2) |
| ----------------------------------------------- |
\                        4                        /