How do I solve a second order non linear differential equation using matlab.

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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Answer 2

Lewis Fer 2021-6-10

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https://ww2.mathworks.cn/matlabcentral/answers/358284-how-do-i-solve-a-second-order-non-linear-differential-equation-using-matlab#answer_721855

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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Answer 3

James Tursa 2017-9-25

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https://ww2.mathworks.cn/matlabcentral/answers/358284-how-do-i-solve-a-second-order-non-linear-differential-equation-using-matlab#answer_283100

编辑：James Tursa 2017-9-25

在 MATLAB Online 中打开

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