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미분방정식풀기

초기조건의유무와관계없이dsolve함수를사용하여해석적으로미분방정식의해를구합니다。연립미분방정식의해를구하려면연립미분방정식풀기항목을참조하십시오。

다음미분방정식을풀어보겠습니다。

$\frac{d y}{d t} ＝ t y ．$

기호 함수y (t)를생성하려면우선信谊를사용하여y를나타내십시오。

信谊y (t)

= =를사용하여방정식을정의하고diff함수를사용하여미분을현합니다。

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

Ode (t) = diff(y(t)， t) == t*y(t)

dsolve를사용하여방정식을풉니다。

ySol(t) = dsolve(ode)

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

이전해법에서는조건을지정하지않았기때문에상수C1이나타납니다。초기 조건Y (0) == 2를사용하여방정식을풉니다。dsolve함수는이조건을충족시키는C1의값을구합니다。

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

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

dsolve로방정식을풀수없다면수치적으로방정식을풀어보십시오。수치적으로2계미분방정식풀기항목을참조하십시오。

초기조건이있는다음비선형미분방정식을풀어보십시오。이방정식에는여러개의해가있습니다。

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

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)

2개의초기조건이있는다음2계미분방정식을풀어보십시오。

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

방정식과조건을정의합니다。두번째초기조건은y의1계도함수를포함합니다。기호 함수Dy = diff(y)를작성하여도함수를현한다음Dy (0) = = 0을사용하여조건을정의합니다。

syms y(x) Dy = diff(y);Ode = diff(y,x,2) == cos(2*x)-y;Cond1 = y(0) == 1;cond2 = Dy(0) == 0;

y에 대해颂歌를풉니다。简化함수를사용하여해를단순화합니다。

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

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

3개의초기조건이있는다음3계미분방정식을풀어보십시오。

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

초기조건에는1계및2계도함수가포함되어있으므로두개의기호함수Du = diff(u,x)및D2u = diff(u,x,2)를작성하여초기조건을지정합니다。

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

방정식과초기조건을작성하고이방정식을풉니다。

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) =(π* exp (x)) / 3 - exp (- x / 2) * cos ((3 ^ (1/2) * x) / 2) *(π/ 3 - 1)-…(3^(1/2)*exp(-x/2)*sin((3^(1/2)*x)/2)*(+ 1))/3

다음표에는여러미분방정식의예와관련符号数学工具箱™구문이나와있습니다。마지막예제는에어리미분방정식이며，이러한방정식의해를에어리함수라고합니다。

미분방정식	MATLAB^®명령
$\begin{array}{l} \frac{d y}{d t} + 4 y （ t ）＝ e^{- t} ， \\ y （ 0 ）＝ 1. \end{array}$	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
$2 x^{2} \frac{d^{2} y}{d x^{2}} + 3. x \frac{d y}{d x} - y ＝ 0.$	信谊y (x)颂歌= 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)
에어리방정식。 $\frac{d^{2} y}{d x^{2}} ＝ x y （ x ）．$	Syms y(x) ode = diff(y,x,2) == xy;ySol(x) = dsolve(ode) ySol(x) = C1airy(0,x) + C2*airy(2,x)

미분방정식

MATLAB^®명령

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

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

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

信谊y (x)颂歌= 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)

에어리방정식。

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

Syms y(x) ode = diff(y,x,2) == x*y;ySol(x) = dsolve(ode)

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