matlab---方程求解

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代数方程符号解

>> syms a b c x>> s = a * x ^ 2 + b * x + c;>> solve(s)ans = -(b + (b^2 - 4*a*c)^(1/2))/(2*a) -(b - (b^2 - 4*a*c)^(1/2))/(2*a)%如果想求指定符号解，将其加入solve中>> b = solve(s, b)b =-(a*x^2 + c)/x

代数方程组中符号解
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>> syms x y alpha>> [x, y] = solve(x^2*y^2, x-y/2-alpha)x = alpha     0y =        0 -2*alpha

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>> s = solve('u^2-v^2=a^2', 'u+v=1', 'a^2-2*a=3');>> m = [s.a, s.u, s.v]m =[ -1, 1,  0]   %每一行包含了方程组的一组解[  3, 5, -4]

齐次方程组求解
齐次线性方程AX=0, rank(A)=r

>> a = [1 1 1 1 -3 -1 1;        1 0 0 0 1 1 0;        -2 0 0 -1 0 -1 -2];>> r = rank(a);>> x = null(a, 'r')            %注意r必须加引号x =                            %列向量为ax=0的一个基本解     0    -1    -1     0    -1     2     1     1     1     0     0     0     0     2     1    -2     0     1     0     0     0     0     1     0     0     0     0     1

非齐次方程组求解
rank(a) = r = n

>> a = [5 0 4 2; 1 -1 2 1; 4 1 2 0; 1 1 1 1];>> b = [3; 1; 1; 0];>> x = zeros(4, 1);     %4元列向量>> x = linsolve(a, b)x =    1.0000   -1.0000   -1.0000    1.0000

rank(a) = r > n

>> a = [2 -1 3;3 1 -5;4 -1 1;1 3 -13];>> b = [3; 0; 3; -6];>> x = a\bx =    1.0000    2.0000    1.0000

微分方程求解

>> dsolve('Dy = t * y')ans =C2*exp(t^2/2)%也可加入初始条件>> x = dsolve('Dy = t * y', 'y(0) = 2')x =2*exp(t^2/2)

微分方程组求解

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>> y = dsolve('D2y=x*y', 'y(0)=0', 'y(3)=besselk(1/3, 2*sqrt(3))/pi', 'x')y =(3^(1/2)*besselk(1/3, 2*3^(1/2))*airy(0, x))/(pi*(3^(1/2)*airy(0, 3) - airy(2, 3))) - (besselk(1/3, 2*3^(1/2))*airy(2, x))/(pi*(3^(1/2)*airy(0, 3) - airy(2, 3)))

求导

>> syms x;>> y = sin(5 * x);>> diff(y)        %一阶导ans =5*cos(5*x) >> diff(y, 2)    %二阶导ans =-25*sin(5*x)

>> syms s t>> f = sin(s * t);>> diff(f, t)     %对t求导ans =s*cos(s*t)

求积分

>> syms x t z alpha>> int(-2 * x / (1 + x^2) ^ 2)     %积分ans =1/(x^2 + 1)>> int(x * log(1 + x), 0 , 1)      %定积分ans =1/4>> int(2 * x, sin(t), 1)          %定积分，积分范围为（sin(t), 1)ans =cos(t)^2

求极限

>> syms h n x;>> limit( (cos(x+h) - cos(x) ) / h, h, 0)  %当h趋向于0时的极限ans =-sin(x)>> limit(x/abs(x), x, 0, 'left')    %左极限ans =-1>> limit(x/abs(x), x, 0, 'right')   %右极限ans =1

求级数

>> syms k;>> symsum(1/2^k, 1, inf)    %1到无穷的和ans =1>> symsum(sin(k*pi)*k, 0, n)   %0到n的和ans =- (exp(-pi*(n + 1)*i)*(2*n + 1)*i)/8 + (exp(pi*(n + 1)*i)*(2*n + 1)*i)/8>> symsum(x^k/sym('k!'), k, 0, inf)  %自变量为k，0到无穷的和ans =exp(x)

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