clear

exact = @(x) (x - pi*sinh(2*x)/sinh(2*pi))/4;
approx3 = @(x) 2/5*sin(x) - 1/8*sin(2*x) + 2/39*sin(3*x);
approx5 = @(x) approx3(x) -1/40*sin(4*x) + 2/145*sin(5*x);

a = 0;
b = pi;
np = 100;
x = linspace(a, b, np);

figure(1)
plot(x, exact(x), '-')
hold on
plot(x, approx3(x), '-')
legend('exact', 'galerkin-3', 'location', 'best')
grid on
xlabel('x')
title('Exact soulution of an ODE vs 3-term Galerkin approximations')

figure(2)
plot(x, exact(x), '-')
hold on
plot(x, approx5(x), '-')
legend('exact', 'galerkin-5', 'location', 'best')
grid on
xlabel('x')
title('Exact soulution of an ODE vs 5-term Galerkin approximations')

nn = 1000;
xx = linspace(a, b, nn);
error3 = norm(exact(xx) - approx3(xx), inf)
error5 = norm(exact(xx) - approx5(xx), inf)