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 个评论
James Tursa
2017-9-25
What have you done so far? What specific problems are you having with your code? Have you looked at the doc for ode45 to see examples of how to use ode45 to numerically solve 2nd order ODE's?
Patrick Guarente
2017-9-25
James Tursa
2017-9-25
编辑：James Tursa
2017-9-25
"... 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.
Patrick Guarente
2017-9-25
回答（4 个）
Teja Muppirala
2017-9-26
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.
Lewis Fer
2021-6-10
1 个投票
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)
James Tursa
2017-9-25
编辑：James Tursa
2017-9-25
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
2017-9-26
According to MATHEMATICA, the analytical solution is
z(x) = log(cosh(sqrt(A*B)*x))/A
Best wishes
Torsten.
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