Second order ordinary differential equation

I am trying to find the exact solution of this differential equation, but the error 'explicit solution not found' occur -y''(x) +2cos2x*y(x) -lambda*y(x) =0

[Dyuman Joshi]

Please show the code that resulted in the "error" mentioned.

[Abdul]

I am using the command dsolve for finding the exact solution of this problem. If you have a code that works, kindly share it thanks

[Walter Roberson]

The notation is a bit ambiguous.

Note that it matters in the end.

syms y(x) lambda
dy = diff(y);
d2y = diff(dy);
eqn = d2y + 2  * cos(2*x) * y - lambda*y == 0

eqn(x) = 

dsolve(eqn)

Warning: Unable to find symbolic solution.

 
ans =
 
[ empty sym ]
 

eqn2 = d2y + 2  * cos(2*x * y) - lambda*y == 0

eqn2(x) = 

dsolve(eqn2)

Warning: Unable to find symbolic solution.

 
ans =
 
[ empty sym ]
 

Answer 1

[Sam Chak]

Hi @Abdul

I believe that 'explicit solution not found' is more of a notification than an error message. Upon closer inspection, your second-order system appears to resemble the Mathieu Differential Equation. If that's the case, the solution is provided in the form of the Mathieu function. For additional information, please refer to the following file on File Exchange:

https://www.mathworks.com/matlabcentral/fileexchange/34381-mathieu-equation-parametric-oscillator

[Sam Chak]

@Abdul, I don't know how to express the Mathieu functions in MATLAB, but I simulated the Mathieu differential equation for different values of lambda (λ) to observe the stability of the solutions.

lambda = 1:6;

t   = 0:0.01:60;
y0  = [1; 0];

for j = 1:numel(lambda)
    sol = ode45(@(t, y) MathieuDE(t, y, lambda(j)), t, y0);
    y   = deval(sol, t);
    subplot(2, 3, j)
    plot(y(1,:), y(2,:)), grid on
    xlabel('y_{1}'), ylabel('y_{2}')
    title("\lambda = "+string(lambda(j)))
    axis equal 
end

%% Mathieu Differential Equation
function dydt = MathieuDE(t, y, lambda)
    dydt    = zeros(2, 1);
    dydt(1) = y(2);
    dydt(2) = 2*cos(2*t)*y(1) - lambda*y(1);
end