Question

Solve by linear differentiable
equation
sec x dy/dx = y + sin x​

Answer 1

Answer:

y + ( 1 + Sinx ) = c e ^ ( Sinx )

Step-by-step explanation:

Given----> Secx dy/dx = y + Sinx

To find-----> Solution of given differential equation

Solution------>

Secx dy / dx = y + Sinx

=> dy / dx = y / Secx + Sinx / Secx

=> dy / dx = y Cosx + Sinx Cosx

=> dy / dx = y Cosx + Sinx Cosx

=> dy / dx - y Cosx = Sinx Cosx

Comparing it with , dy / dx + Py = Q , we get,

P = - Cosx , Q = Sinx Cosx

∫ P dx

I. F. = e

∫ - Cosx dx

= e

- Sinx

= e

Now , solution is ,

- Sinx - Sinx

y e = ∫ Sinx Cosx e dx + C

Let,

- Sinx = t

=> - Cosx dx = dt

- Sinx

=> y e = - ∫ -t eᵗ dt + C

= ∫ t eᵗ dt + C

Applying intregation by parts , we get,

= t eᵗ - ∫ 1 eᵗ dt + C

= t eᵗ - eᵗ + c

= eᵗ ( t - 1 ) + C

- Sinx - Sinx

y e = e ( - Sinx - 1 ) + C

Sinx

y = - ( 1 + Sinx ) + C e

y + ( 1 + Sinx ) = C e^ ( Sinx )