在八度中，是否有办法为一个变量求解两个变量中的方程

Question

我有一个MATLAB脚本，它使用拉普拉斯变换解决了非均匀的一阶线性IVP。 （在这个例子中，脚本设置为解决IVP ，）

syms x(t) s X; 

a0 = -3; 
x0 = 4; 
rhs = t^2; 

lhs = diff(x,t) + a0*x; 
ode = lhs - rhs 

Lx = X; 
LDx = s*X - x0; 
LHS = LDx + a0*Lx; 
RHS = laplace(rhs,t,s); 
IVP = LHS - RHS; 

IVP = collect(IVP,X); 

X = solve(IVP, X); 
X = partfrac(X); 

sol = ilaplace(X, s, t) 
check1 = diff(sol,t) - 3*sol 
check2 = vpa(subs(sol, t, 0)) 

如果我取代“因素”为“收集”，剧本上八度几乎作品与象征打包链接到SymPy，除了“解决”命令https://www.mathworks.com/help/symbolic/solve.html。

是否有任何八度（或SymPy，如果这将起到解决方法的作用）命令将作为一个MATLAB符号工具箱“解决”命令，所以我可以通过带有脚本的拉普拉斯变换来解决IVP，不得不手动解决X，然后使用“ilaplace”？

在此先感谢您提供的任何帮助。

Answer 1

好的，我的一个学生解决了这个问题（我会在本周晚些时候联系他，看他是否希望公开承认他的解决方案）。

你只需要定义coeff*[1;X]结果作为设置为0的公式，说IVPEQ = coeff*[1;X] == 0，然后使用符号包命令solve在这个公式中，X = solve(IVPEQ, X)。

这里是一个版本，我以前的一阶IVP解算器与我的学生的修改

syms x(t) s X; 

a0 = -3; 
x0 = 4; 
rhs = t^2; 

lhs = diff(x,t) + a0*x; 
ode = lhs - rhs 

Lx = X; 
LDx = s*X - x0; 
LHS = LDx + a0*Lx; 
RHS = laplace(rhs,t,s); % The t and s in laplace aren't necessary, as they are default 
IVP = LHS - RHS; 

coeff = coeffs(IVP,X); 
IVPEQ = coeff*[1;X] == 0; 

X = solve(IVPEQ,X); 

X = partfrac(X); 

sol = ilaplace(X, s, t) 

Dsol = diff(sol,t); 
check1 = Dsol + a0*sol 
check2 = vpa(subs(sol, t, 0)) 

，这里是第二次IVP解算器与学生的修改再次

syms x(t) s X; 

a1 =-2; 
a0 = -3; 
x0 = 4; 
xdot0 = 5; 
rhs = t^2; 

Dx = diff(x,t); 
D2x = diff(x,t,2); 
lhs = D2x + a1*Dx + a0*x; 
ode = lhs - rhs 

Lx = X ; 
LDx = s*X - x0; 
LD2x = s^2*X - x0*s - xdot0; 
LHS = LD2x + a1*LDx + a0*Lx; 
RHS = laplace(rhs,t,s); % The t and s in laplace aren't necessary, as they are default 
IVP = LHS - RHS; 

coeff = coeffs(IVP,X); 
IVPEQ = coeff*[1;X] == 0; 

X = solve(IVPEQ,X); 

X = partfrac(X); 

sol = ilaplace(X, s, t) 

Dsol = diff(sol,t); 
D2sol = diff(sol,t,2); 
check1 = D2sol + a1*Dsol + a0*sol 
check2 = vpa(subs(sol, t, 0)) 
check3 = vpa(subs(Dsol, t, 0)) 

谢谢， @Tasos_Papastylianou，为您提供巨大的帮助！

Answer 2

以下是一些Octave脚本（至少大概）重现上述MATLAB脚本。您必须按照X脚本手动输入IVP = 0的解决方案到每个脚本的第2部分，但它们确实起作用。如果任何人有无法像MATLAB解决函数那样根据X解决Octave求解IVP = 0，我很乐意听到它。

这一对解决了$ \ dot {x} - 3x = t^2 $，$ x（0）= 4 $。

1部分：

syms x(t) s X; 

a0 = -3; 
x0 = 4; 
rhs = t^2; 

lhs = diff(x,t) + a0*x; 
ode = lhs - rhs 

Lx = X; 
LDx = s*X - x0; 
LHS = LDx + a0*Lx; 
RHS = laplace(rhs,t,s); % The t and s in laplace aren't necessary, as they are default 
IVP = LHS - RHS; 

coeff = coeffs(IVP,X); 
IVP = coeff*[1;X] 

第2部分：

syms x(t) s X; 

X = -1*((-4*s^3-2)/s^3)/(s-3) 

X = partfrac(X); 

sol = ilaplace(X, s, t) 
check1 = diff(sol,t) - 3*sol 
check2 = vpa(subs(sol, t, 0)) 

这对解决了$ \ DDOT {X} - 2 \点{X} - 3×= T^2个，$ X （0）= 4 $，$ \ dot {x}（0）= 5 $。

1部分：

syms x(t) s X; 

a1 =-2; 
a0 = -3; 
x0 = 4; 
xdot0 = 5; 
rhs = t^2; 

Dx = diff(x,t); 
D2x = diff(x,t,2); 
lhs = D2x + a1*Dx + a0*x; 
ode = lhs - rhs 

Lx = X ; 
LDx = s*X - x0; 
LD2x = s^2*X - x0*s - xdot0; 
LHS = LD2x + a1*LDx + a0*Lx; 
RHS = laplace(rhs,t,s); % The t and s in laplace aren't necessary, as they are default 
IVP = LHS - RHS; 

coeff = coeffs(IVP,X); 
IVP = coeff*[1;X] 

第2部分：

syms x(t) s X; 

a1 = -2; 
a0 = -3; 

X = -1*((-4*s^4 + 3*s^3 - 2)/s^3)/(s^2 - 2*s - 3) 

X = partfrac(X); 

sol = ilaplace(X, s, t) 

Dsol = diff(sol,t); 
D2sol = diff(sol,t,2); 
check1 = D2sol + a1*Dsol + a0*sol 
check2 = vpa(subs(sol, t, 0)) 
check3 = vpa(subs(Dsol, t, 0)) 

非常感谢所有帮助和建议！这是真正的赞赏！