在八度中，是否有辦法爲一個變量求解兩個變量中的方程

Question

我有一個MATLAB腳本，它使用拉普拉斯變換解決了非均勻的一階線性IVP。 （在這個例子中，腳本設置爲解決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」爲「收集」，劇本上八度幾乎作品與象徵打包鏈接到SymPy，除了「解決」命令https://www.mathworks.com/help/symbolic/solve.html。

是否有任何八度（或SymPy，如果這將起到解決方法的作用）命令將作爲一個MATLAB符號工具箱「解決」命令，所以我可以通過帶有腳本的拉普拉斯變換來解決IVP，不得不手動解決X，然後使用「ilaplace」？

在此先感謝您提供的任何幫助。

Answer 1

好的，我的一個學生解決了這個問題（我會在本週晚些時候聯繫他，看他是否希望公開承認他的解決方案）。

你只需要定義coeff*[1;X]結果作爲設置爲0的公式，說IVPEQ = coeff*[1;X] == 0，然後使用符號包命令solve在這個公式中，X = solve(IVPEQ, X)。

這裏是一個版本，我以前的一階IVP解算器與我的學生的修改

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)) 

，這裏是第二次IVP解算器與學生的修改再次

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)) 

謝謝， @Tasos_Papastylianou，爲您提供巨大的幫助！

Answer 2

以下是一些Octave腳本（至少大概）重現上述MATLAB腳本。您必須按照X腳本手動輸入IVP = 0的解決方案到每個腳本的第2部分，但它們確實起作用。如果任何人有無法像MATLAB解決函數那樣根據X解決Octave求解IVP = 0，我很樂意聽到它。

這一對解決了$ \ 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)) 

這對解決了$ \ DDOT {X} - 2 \點{X} - 3×= T^2個，$ 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)) 

非常感謝所有幫助和建議！這是真正的讚賞！