Question

If φ(x) is a differentiable function, then the solution of the differential equation dy+yφ'(x)-φ(x)φ'(x)dx=0 is

• A. \( y=\{\phi(x)-1\}+Ce^{-\phi(x)} \)
• B. \( y\phi(x)=\{\phi(x)\}^2+C \)
• C. \( ye^{\phi(x)}=\phi(x)e^{\phi(x)}+C \)
• D. y-φ(x)=φ(x)e⁻φ(x)

Hint:

Identify linear differential equations and use integrating factor.

Answer 1

The Correct Option is C

dy + yφ'(x)dx = φ(x)φ'(x)dx This is a linear differential equation with integrating factor: IF=eⁱⁿᵗ φ'⁽ˣ⁾ᵈˣ=e^φ(x) ⟹ (d)/(dx)(ye^φ(x))=φ(x)φ'(x)e^φ(x) Integrating, ye^φ(x)=φ(x)e^φ(x)+C