var('x, x1, x2, t')

# Finding a general solution
print("General solution:")
y = function('y', x)
ode = 4*diff(y, x, 2) + 36 * y == csc(3 * x)
y(x) = desolve(ode, y)
y.show()

# Solving an initial-value problem
print("Solving an initial-value problem:")
y = function('y', x)
ode = diff(y, x, 2) - 4 * diff(y, x) + 13 * y == 0
y(x) = desolve(ode, y, [0, -1, 2], ivar=x)
y.show()

# Solving a system of first-order equations
print("Solving a system of first-order ODEs:")
x1 = function('x1', t)
x2 = function('x2', t)
ode1 = 2 * diff(x1, t) + diff(x2, t) - x2 == t
ode2 = diff(x1, t) + diff(x2, t) == t^2
y1, y2 = desolve_system([ode1, ode2], [x1, x2], 
    ics=[0, 0, 0])
y1.show()
y2.show()