what's wrong with matlab dsolve ?

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 ?

Answer 1

[Star Strider]

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 = 

sl_rw = rewrite(sl, 'sinhcosh')

sl_rw = 

sl_rw = rewrite(sl, 'sinh')

sl_rw = 

sl_rw = simplify(sl_rw, 500)

sl_rw = 

.

Answer 2

[Torsten]

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) )

sol = 

[Dario]

thank you this makes sense now.

Answer 3

[Sam Chak]

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