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);
subplot(2,1,1)
plot(sol.x,sol.y(1,:),'r','lineWidth',2)
xlim([0.9998 1])
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
Best Answer