- 求解常微分方程的通解
- 求解常微分方程的初边值问题
- 求解常微分方程组：
试解常微分方程：
x
2
+
y
+
(
x
−
2
y
)
y
′
=
0
x^2+y+left( x-2y right) y^{'}=0
x2+y+(x−2y)y′=0
尝试用手算了一下，没算出来。。。
syms y(x); f=x^2+y+(x-2*y)*diff(y) dsolve(f==0)
结果为：
x
±
4
x
3
3
+
x
2
+
C
1
2
frac{xpm sqrt{frac{4x^3}{3}+x^2+C_1}}{2}
2x±34x3+x2+C1
y ′ ′ ′ − y ′ ′ = x , y ′ ( 1 ) = 8 , y ′ ′ ( 2 ) = 4 y^{'''}-y^{''}=x,y^{'}left( 1 right) =8,y^{''}left( 2 right) =4 y′′′−y′′=x,y′(1)=8,y′′(2)=4
syms y(x); dy=diff(y);d2y=diff(y,2) f=diff(y,3)-diff(y,2)-x y=dsolve(f==0,dy(1)==7,d2y(2)==4) simplify(y)
simplify(y)可以简化结果
结果为：
7
e
x
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2
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7
e
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17
2
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x
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x
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2
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6
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1
6
7e^{x-2}-left( frac{7}{e}-frac{17}{2} right) x-frac{x^2}{2}-frac{x^3}{6}+frac{1}{6}
7ex−2−(e7−217)x−2x2−6x3+61
{ f ′ ′ + 3 g = sin x g ′ + f ′ = cos x 通解和初边值条件为 f ′ ( 2 ) = 0 , f ( 3 ) = 3 , g ( 5 ) = 1 的解 begin{cases} f^{''}+3g=sin x\ g^{'}+f^{'}=cos x\ end{cases}text{通解和初边值条件为}f^{'}left( 2 right) =0,fleft( 3 right) =3,gleft( 5 right) =1text{的解} {f′′+3g=sinxg′+f′=cosx通解和初边值条件为f′(2)=0,f(3)=3,g(5)=1的解
% 求解常微分方程组
syms f(x) g(x);
df=diff(f)
[f1,g1]=dsolve(diff(f,2)+3*g==sin(x),diff(g)+df==sin(x))
f1=simplify(f1),g1=simplify(g1)
% 带有边值条件
[f2,g2]=dsolve(diff(f,2)+3*g==sin(x),diff(g)+df==sin(x),df(2)==0 ,...
f(3)==3,g(5)==1)
f2=simplify(f2),g2=simplify(g2)
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