How to solve "system contains a nonlinear equation"

Hello, when I run the code below, I get an error because the square of the second derivative of y(x) is used. Could someone please tell me how I can solve this type of problem?

syms y(x) x Y 
mum=2;
r=0.05;
m=0.01;
p=1;
s=1;
t=0.1;
a=0.25;
b=0.25;
l=0.5;
t0= ((1/(r+p*s-m))^(1/(1-a-b)))+((1/(r+p*s-m))^(1/(1-a-b)))*((1/(r+t*p*s-m))^(1/(1-l)));
Dy = diff(y);
D2y = diff(y,2);
mo=1/(r-((p*Dy*x*s*mum)/y)-((Dy*x*mum)/y)+((Dy*x*mum*m)/y)-((Dy*x*(s^2)*(mum^2))/y)-((D2y*(x^2)*(s^2)*(mum^2))/(2*y)));
mi=1/(r-((t*p*Dy*x*s*mum)/y)-((Dy*x*mum)/y)+((Dy*x*mum*m)/y)-((Dy*x*(s^2)*(mum^2))/y)-((D2y*(x^2)*(s^2)*(mum^2))/(2*y)));
ode = y-(1/x)*(mo^(1/(1-a-b))+mo^(1/(1-a-b))* mi^(1/(1-l)));
[VF,Subs] = odeToVectorField(ode);

Error using mupadengine/feval_internal
System contains a nonlinear equation in 'diff(y(x), x, x)'. The system must be quasi-linear: highest derivatives must enter the differential equations linearly.

Error in odeToVectorField>mupadOdeToVectorField (line 171)
T = feval_internal(symengine,'symobj::odeToVectorField',sys,x,stringInput);

Error in odeToVectorField (line 119)
sol = mupadOdeToVectorField(varargin);

odefcn = matlabFunction(VF, 'Vars',{x,Y});
bcfcn = @(ya,yb)[ya(1)-t0;yb(1)];
xmesh = linspace(1,80,150);
solinit = bvpinit(xmesh, [0 0]);
sol = bvp4c(odefcn,bcfcn,solinit);
plot(sol.x,sol.y(1,:))

Answer 1

[Walter Roberson]

syms y(x) x Y 
mum=2;
r=0.05;
m=0.01;
p=1;
s=1;
t=0.1;
a=0.25;
b=0.25;
l=0.5;
t0= ((1/(r+p*s-m))^(1/(1-a-b)))+((1/(r+p*s-m))^(1/(1-a-b)))*((1/(r+t*p*s-m))^(1/(1-l)));
Dy = diff(y);
D2y = diff(y,2);
mo=1/(r-((p*Dy*x*s*mum)/y)-((Dy*x*mum)/y)+((Dy*x*mum*m)/y)-((Dy*x*(s^2)*(mum^2))/y)-((D2y*(x^2)*(s^2)*(mum^2))/(2*y)));
mi=1/(r-((t*p*Dy*x*s*mum)/y)-((Dy*x*mum)/y)+((Dy*x*mum*m)/y)-((Dy*x*(s^2)*(mum^2))/y)-((D2y*(x^2)*(s^2)*(mum^2))/(2*y)));
ode = y-(1/x)*(mo^(1/(1-a-b))+mo^(1/(1-a-b))* mi^(1/(1-l)));

ode

ode(x) = 

The denominator has d^2y/dx^2 in it, and that term is squared, so the square of d^2y/dx^2 appears in the equation.

sode = simplify(expand(ode(x)), 'steps', 50)

sode = 

If you look at which is d^2y/dx^2 you can see that it appears to the 4th power in the expanded version, so the situation might be even worse than it seems at first.

[Walter Roberson]

You can get rid of that particular issue by doing an explicit expansion of a 4th order root:

syms y(x) x Y 
mum=2;
r=0.05;
m=0.01;
p=1;
s=1;
t=0.1;
a=0.25;
b=0.25;
l=0.5;
t0= ((1/(r+p*s-m))^(1/(1-a-b)))+((1/(r+p*s-m))^(1/(1-a-b)))*((1/(r+t*p*s-m))^(1/(1-l)));
Dy = diff(y);
D2y = diff(y,2);
mo=1/(r-((p*Dy*x*s*mum)/y)-((Dy*x*mum)/y)+((Dy*x*mum*m)/y)-((Dy*x*(s^2)*(mum^2))/y)-((D2y*(x^2)*(s^2)*(mum^2))/(2*y)));
mi=1/(r-((t*p*Dy*x*s*mum)/y)-((Dy*x*mum)/y)+((Dy*x*mum*m)/y)-((Dy*x*(s^2)*(mum^2))/y)-((D2y*(x^2)*(s^2)*(mum^2))/(2*y)));
eqn = y == (1/x)*(mo^(1/(1-a-b))+mo^(1/(1-a-b))* mi^(1/(1-l)));

ode = isolate(eqn, D2y)

ode = 

R4 = findSymType(ode, 'root');
solR4 = solve(children(R4, 1), children(R4,2), 'MaxDegree', 4);
newode = subs(ode, R4, solR4(1))

newode = 

snew = isolate(simplify(newode, 'steps', 50), D2y)

snew = 

findSymType(snew, 'diff')

ans = 

[VF,Subs] = odeToVectorField(snew);

Error using mupadengine/feval_internal
System does not seem to contain a first-order differential equation for the field 'D(y)(t)'.

Error in odeToVectorField>mupadOdeToVectorField (line 171)
T = feval_internal(symengine,'symobj::odeToVectorField',sys,x,stringInput);

Error in odeToVectorField (line 119)
sol = mupadOdeToVectorField(varargin);

... but you still have problems.

[Fatemeh]

Thanks so much @Walter Roberson. I see your point. What if I change the code as follows? I now have the square of y(x)'s second derivative. So, what should I do in this case?

syms y(x) x Y 
mum=2;
r=0.05;
m=0.01;
p=1;
s=1;
t=0.1;
a=0.25;
b=0.25;
l=0.5;
t0= ((1/(r+p*s-m))^(1/(1-a-b)))+((1/(r+p*s-m))^(1/(1-a-b)))*((1/(r+t*p*s-m))^(1/(1-l)));
Dy = diff(y);
D2y = diff(y,2);
mo=1/(r-((p*Dy*x*s*mum)/y)-((Dy*x*mum)/y)+((Dy*x*mum*m)/y)-((Dy*x*(s^2)*(mum^2))/y)-((D2y*(x^2)*(s^2)*(mum^2))/(2*y)));
mi=1/(r-((t*p*Dy*x*s*mum)/y)-((Dy*x*mum)/y)+((Dy*x*mum*m)/y)-((Dy*x*(s^2)*(mum^2))/y)-((D2y*(x^2)*(s^2)*(mum^2))/(2*y)));
ode = y-(1/x)*(mo^(1/(1-a-b))+ mi^(1/(1-l)));
[VF,Subs] = odeToVectorField(ode);

Error using mupadengine/feval_internal
System contains a nonlinear equation in 'diff(y(x), x, x)'. The system must be quasi-linear: highest derivatives must enter the differential equations linearly.

Error in odeToVectorField>mupadOdeToVectorField (line 171)
T = feval_internal(symengine,'symobj::odeToVectorField',sys,x,stringInput);

Error in odeToVectorField (line 119)
sol = mupadOdeToVectorField(varargin);

[Torsten]

@Fatemeh

Can you decide which solution for D2y is the "correct" one in your case ?

syms y Dy D2y x Y 
mum=2;
r=0.05;
m=0.01;
p=1;
s=1;
t=0.1;
a=0.25;
b=0.25;
l=0.5;
t0= ((1/(r+p*s-m))^(1/(1-a-b)))+((1/(r+p*s-m))^(1/(1-a-b)))*((1/(r+t*p*s-m))^(1/(1-l)));
mo=1/(r-((p*Dy*x*s*mum)/y)-((Dy*x*mum)/y)+((Dy*x*mum*m)/y)-((Dy*x*(s^2)*(mum^2))/y)-((D2y*(x^2)*(s^2)*(mum^2))/(2*y)));
mi=1/(r-((t*p*Dy*x*s*mum)/y)-((Dy*x*mum)/y)+((Dy*x*mum*m)/y)-((Dy*x*(s^2)*(mum^2))/y)-((D2y*(x^2)*(s^2)*(mum^2))/(2*y)));
ode = y-(1/x)*(mo^(1/(1-a-b))+ mi^(1/(1-l)))

ode = 

solve(ode==0,D2y)

ans = 

[Fatemeh]

The second one.

[Torsten]

Then try it out:

syms y Dy D2y x Y 
mum=2;
r=0.05;
m=0.01;
p=1;
s=1;
t=0.1;
a=0.25;
b=0.25;
l=0.5;
t0= ((1/(r+p*s-m))^(1/(1-a-b)))+((1/(r+p*s-m))^(1/(1-a-b)))*((1/(r+t*p*s-m))^(1/(1-l)));
mo=1/(r-((p*Dy*x*s*mum)/y)-((Dy*x*mum)/y)+((Dy*x*mum*m)/y)-((Dy*x*(s^2)*(mum^2))/y)-((D2y*(x^2)*(s^2)*(mum^2))/(2*y)));
mi=1/(r-((t*p*Dy*x*s*mum)/y)-((Dy*x*mum)/y)+((Dy*x*mum*m)/y)-((Dy*x*(s^2)*(mum^2))/y)-((D2y*(x^2)*(s^2)*(mum^2))/(2*y)));
ode = y-(1/x)*(mo^(1/(1-a-b))+ mi^(1/(1-l)))

ode = 

D2ynum = solve(ode==0,D2y);
D2ynum = D2ynum(2);
f = matlabFunction(D2ynum,"Vars",{x, [y Dy]});
odefcn = @(x,y)[y(2);f(x,[y(1),y(2)])];
bcfcn = @(ya,yb)[ya(1)-t0;yb(1)];
xmesh = linspace(1,80,150);
solinit = bvpinit(xmesh, [1 1]);
sol = bvp4c(odefcn,bcfcn,solinit);
plot(sol.x,sol.y)

[Fatemeh]

Thanks much @Torsten