sl =
what's wrong with matlab dsolve ?
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I am using this matlab version
'24.2.0.2806996 (R2024b) Update
Example:
-------------------------------------------
syms u(t) a
sl = dsolve( diff(u,t) == sqrt( u^2+a^2) )
sl =
-a*1i
a*1i
exp(C1 + t)/2 - (a^2*exp(- C1 - t))/2
------------------------------------------
Apart from the constant solution,
the correct solution is
u(t) = a*sinh( t+ const.)
which is different than the matlab solution unless a^2 = 1
What's wrong with dsolve ?
回答（3 个）
You can use the rewrite function to get it in a particular form, however MATLAB doesn't appear to agree with your solution --
syms u(t) a
sl = dsolve( diff(u,t) == sqrt( u^2+a^2) )
sl_rw = rewrite(sl, 'sinhcosh')
sl_rw = rewrite(sl, 'sinh')
sl_rw = simplify(sl_rw, 500)
.
If a > 0, insert C1 = const + log(a) in MATLAB's solution for the free constant C1, and you will recover your solution.
For a < 0, your solution won't work because you get
a*cosh( t+ const ) = abs(a)*cosh(t+ const)
from
diff(u,t) = sqrt(a^2+u^2)
for
u(t) = a*sinh(t + const)
which is only valid if a >= 0.
For a = 0, your solution is only u = 0 whereas MATLAB gives u = 0 and u = exp(C1 + t)/2 (or simpler u = const*exp(t) for a free constant const >= 0) which is correct.
So this will work:
syms a positive
syms u(t)
sol = dsolve( diff(u,t) == sqrt( u^2+a^2) )
Hi @Dario
This is not a solution to your problem but a demo to show the comparison between the numerical solution and the analytical solution, given that the parameter a is a real number that can take on either positive or negative values. However, I haven't tested whether dsolve can achieve this level of simplification after applying the simplify() and rewrite() functions.
Differential equation:
Analytical solution (real):
a = -3; % parameter
ode = @(t, u) sqrt(u^2 + a^2);
tspan = [0, 3]; % duration
u0 = 2; % initial value
% ode45 solution
[t, u] = ode45(ode, tspan, u0);
% analytical solution
tsol = linspace(0, tspan(2), 21);
usol = abs(a)*sinh(tsol + asinh(u0/abs(a)));
% result
hold on
plot(t, u)
plot(tsol, usol, '.');
xlabel('t')
ylabel('u(t)')
legend('ode45 solution', 'analytical solution', 'location', 'northwest', 'fontsize', 14)
hold off
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2026-3-17
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