Can you solve the following ODE?

Question

dy/dx = 5*sin(√(17)x+y)

Answer 1

Put √(17)x + y = t

Now differentiating both the sides we get,

√17 + dy/dx = dt/dx

Noy dy/dx = dt/dx - √17, substitute dy/dx in your differential equation we get,

dt/dx = 5sint + √17

Now simply integrate by variable seperation method,

dt / (5sint + √17) = dx, now integrating both the sides we get, =

(arctanh((5 + √17) * tan( t / 2 )) / (2 * √2)) / √2) = -x + Constant

Now substitute the value of t here we will get the answer,

(arctanh((5 + √17) * tan( (√(17)x + y) / 2 )) / (2 * √2)) / √2) = -x + Constant

(arctanh is inverse hyperbolic function)

Answer 2

Let me write a for rt17.

Assume ax+y =u so a + dy/dx = du/dx.

Now the eqn changes to du/dx - 1 = 5sinu

Or du/(1+5sinu) = dx.

Now integrate. You will need, sin2T = [1 - (tanT)^2]/[1 + (tanT)^2], to be used.