usol(x) = [ empty sym ] for using dsolve with the following nonlinear 3rd oder ODE

I am trying to solve this 3rd order nonlinear ode using symbolic solver and it returned usol(x) = [ empty sym ]. Can someone help me understand what is going on with this? I have included my code below. Thanks in advance.

syms u(x)

Du=diff(u,x);
D2u=diff(u,x,2);

ode=diff(u,x,3)+0.5*diff(u,x,2)*u==0;
cond1=u(0)==0;
cond2=Du(0)==0;
cond3=D2u(0)==1;
conds = [cond1 cond2 cond3];

usol(x)=dsolve(ode,conds)

Answer 1

[Walter Roberson]

There is no solution for that. The u(0)=0 condition forces u to be identical to 0. If you instead use the D2u(0)=1 condition, then that by itself forces u(x) = 6/(x+12^(1/3)) which can only reach 0 at x = +/- infinity.

[YX Sun]

Could you please explain what you mean u(0)=0 will forces u to be identical to 0? I used ode45 and it was able to solve it numerically. It is not identically 0 with all three initial conditions imposed. Thanks again.

[Walter Roberson]

Differential equations can be solved numerically or as equations, creating the function for which the differential statements and the boundary conditions hold.

The combination

diff(u,x,3)+0.5*diff(u,x,2)*u==0
u(0)==0

when solved as equations, has only the solution u(x) = 0. That solution is consistent with u(0) == 0 and with Du(0) == 0, but it is not consistent with D2u(0) == 1

At the moment I do not have a description handy for what happens when you do a numeric solution to differential equations such as using ode45(). I will try to come up with some wording later.

[YX Sun]

Thank you Walter. Hope to hear more about this soon.