How to solve a double complex ode using ode45 in MATLAB

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嘉杰 程
嘉杰 程 2021년 10월 16일
댓글: 嘉杰 程 2021년 10월 17일
The odes are:
ode(1) = diff(x,t) == (sqrt(1+x^2)*y)/(x*(t^2));
ode(2) = diff(y,t) == (x^3)*(t^2);
given x(0)=infty; y(0)=0.
I want to draw two picture to reflect x^3(u)-y relation and the u axis should be drawn by log(u) scale.

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Walter Roberson
Walter Roberson 2021년 10월 16일
The infinite boundary condition is a problem for symbolic solution: dsolve() rejects it.
The 0 initial time is a problem: you divide by time, so you generate a numeric infinity at the initial time, which numeric solvers cannot recover from.
If you set initial time above 0, and set the boundary condition to be finite, then you get a singularity. The time of singularity depends upon the boundary condition you set -- with the failure time being approximately 1 over the boundary condition.
syms x(t) y(t)
dx = diff(x)
dx(t) = 
dy = diff(y)
dy(t) = 
odes = [dx == (sqrt(1+x^2)*y)/(x*(t^2));
dy == (x^3)*(t^2)]
odes(t) = 
ic = [x(0) == 1e8, y(0) == 0]
ic = 
sol = dsolve(odes, ic)
Warning: Unable to find symbolic solution.
sol = [ empty sym ]
[eqs,vars] = reduceDifferentialOrder(odes,[x(t), y(t)])
eqs = 
vars = 
[M,F] = massMatrixForm(eqs,vars)
M = 
F = 
f = M\F
f = 
odefun = odeFunction(f,vars)
odefun = function_handle with value:
@(t,in2)[(1.0./t.^2.*in2(2,:).*sqrt(in2(1,:).^2+1.0))./in2(1,:);t.^2.*in2(1,:).^3]
xy0 = double(rhs(ic))
xy0 = 1×2
100000000 0
tspan = [1e-50 100]
tspan = 1×2
0.0000 100.0000
[t, xy] = ode45(odefun, tspan, xy0);
Warning: Failure at t=2.575086e-08. Unable to meet integration tolerances without reducing the step size below the smallest value allowed (5.293956e-23) at time t.
figure
semilogy(t, xy(:,1), 'k+-')
title('x')
figure
plot(t, xy(:,2), 'k*-')
title('y')
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嘉杰 程
嘉杰 程 2021년 10월 17일
thank you!

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추가 답변(1개)

Cris LaPierre
Cris LaPierre 2021년 10월 16일
You can solve a symbolic ODE with initial/boundary conditions using dsolve

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