How do I solve a second order non linear differential equation using matlab.
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I have a fluid dynamics problem and I need to derive an equation for motion.
After applying Newtons second law to the system, and replaceing all the constants with A and B. My equation looks like this.
z'' + A(z')^2 = B
With A and B both being constants.
Initial conditions being that z(0)=0, and z'(0)=0
And I need to solve for z(t).
Thank you
4 comentarios
James Tursa
el 25 de Sept. de 2017
Editada: James Tursa
el 25 de Sept. de 2017
"... I need to find the equation for all time ..."
Are you looking for an analytical/symbolic solution? I thought that you simply wanted a numerical solution given your initial starting values.
Respuestas (4)
Teja Muppirala
el 26 de Sept. de 2017
syms z(t) t A B
zp = diff(z,t);
zpp = diff(z,t,2);
eqn = ( zpp + A*zp^2 == B );
cond = [z(0)==0, zp(0)==0];
zSol = dsolve(eqn,cond,'IgnoreAnalyticConstraints',true);
zSol = unique(simplify(zSol));
This gives 3 solutions:
zSol =
log((C15*sinh(A^(1/2)*B^(1/2)*(t + A*B^(1/2)*1i)))/B^(1/2))/A
log(-(C18*sinh(A^(3/2)*B*1i - A^(1/2)*B^(1/2)*t))/B^(1/2))/A
log(cosh(A^(1/2)*B^(1/2)*t))/A
The first two look weird, but are valid solutions involving complex-valued z. The 3rd solution is real, and that's probably the one that you are looking for.
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Lewis Fer
el 10 de Jun. de 2021
Hello, I am having troubles solving a system of second order nonlinear equations with boundary conditions using MATALB
Here is the equations:
f''(t)=3*f(t)*g(t) -g(t)+5*t;
g''(t)=-4f(t)*g(t)+f(t)-7*t;
the boundary conditions are: f'(0)=0 et h'(o)=5;
g(0)=3 et h'(2)=h(2)
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James Tursa
el 25 de Sept. de 2017
Editada: James Tursa
el 25 de Sept. de 2017
Define a 2-element vector y:
y(1) = z
y(2) = z'
then solve your 2nd order ODE for the highest derivative:
z'' + A(z')^2 = B ==>
z'' = - A(z')^2 + B
then calculate the y element derivative equations, using this z derivative info:
d y(1) = d z = z' = y(2)
d y(2) = d z' = z'' = -A(z')^2 + B = -A*y(2) + B
So create a derivative function based on those two equations, using the function signature that you will find in the ode45 doc. Then call it using the outline provided in the example in the doc.
EDIT: SYMBOLIC SOLUTION
>> dsolve('D2z + A*(Dz)^2 = B')
ans =
C29 + (B^(1/2)*t)/A^(1/2)
C27 - (B^(1/2)*t)/A^(1/2)
log((exp(2*A^(3/2)*B^(1/2)*(C24 + t/A)) - 1)/(2*B^(1/2)*exp(A*(C16 + A^(1/2)*B^(1/2)*(C24 + t/A)))))/A
log((exp(2*A^(3/2)*B^(1/2)*(C20 - t/A)) - 1)/(2*B^(1/2)*exp(A*(C16 + A^(1/2)*B^(1/2)*(C20 - t/A)))))/A
Torsten
el 26 de Sept. de 2017
According to MATHEMATICA, the analytical solution is
z(x) = log(cosh(sqrt(A*B)*x))/A
Best wishes
Torsten.
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