Integrating second order differential equation BVP error

I get the error "Warning: Explicit solution could not be found." when trying to integrate this BVP.

I've attached a picture of the DE here.

https://imgur.com/a/j6eJERx

Why is it failing?

syms y(x)
epsilon = .0001
ode = epsilon*diff(y,x,2)-(3*x+2)*diff(y,x)+y^2 == 0;
cond1 = y(0) == -3/log(2);
cond2 = y(1) == 1;
conds = [cond1 cond2];
ySol(x) = dsolve(ode,conds)
end

Answer 1

[Stephan]

Hi,

i dont think that there is a analytical solution to this problem. You can solve this numeric by using symbolic Toolbox for creating a function handle to this in the first step (or you do this by hand):

syms y(x)
epsilon = .0001
ode = odeToVectorField((epsilon*diff(y,x,2)-(3*x+2)*diff(y,x)+y^2 == 0),y(x));
ode_fun = matlabFunction(ode,'Vars',{'x','Y'})

The code above gives you a system of first order odes and a function handle:

ode =

                                  Y[2]
   10000*(3*x + 2)*Y[2] - 10000*Y[1]^2

vars =

    y
   Dy

ode_fun =

    function_handle with value:

      @(x,Y)[Y(2);Y(1).^2.*-1.0e4+(x.*3.0+2.0).*Y(2).*1.0e4]

Using the vars Information you know that y=Y(1) and Dy=Y(2). Then you can formulate the problem for numeric solving by bvp5c:

odefun = @(x,Y)[Y(2);Y(1).^2.*-1.0e4+(x.*3.0+2.0).*Y(2).*1.0e4];
bcfun = @(ya,yb)[ya(1); yb(1)-1];

x = linspace(0,1,10);
yinit = [0 0];
solinit = bvpinit(x,yinit); 

sol = bvp5c(odefun,bcfun,solinit);

% plot function graph of y
subplot(2,1,1)
plot(sol.x,sol.y(1,:),'r','lineWidth',2)
xlim([0.9998 1])

% plot first derivative of y
subplot(2,1,2)
plot(sol.x,sol.y(2,:),'g','lineWidth',2)
xlim([0.9998 1])

Note that the x-values of the plot are not from 0...1, due to the function would look like a step function in x=1 otherwise.

Best regards

Stephan