The solution of the differential equation $\log_e(\frac{dy}{dx})=3x+4y$ is given by
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → $4e^{3x}+3e^{-4y}+ C = 0$, where C is constant of integration
Given:
$\log\left(\frac{dy}{dx}\right)=3x+4y$
Exponentiate both sides:
$\frac{dy}{dx}=e^{3x+4y}=e^{3x}e^{4y}$
Separate variables:
$e^{-4y}\,dy=e^{3x}\,dx$
Integrate:
$\int e^{-4y}\,dy=\int e^{3x}\,dx$
$-\frac{1}{4}e^{-4y}=\frac{1}{3}e^{3x}+C$
Multiply by 12 to simplify:
$-3e^{-4y}=4e^{3x}+12C$
Rewrite constant: $12C = C'$
$4e^{3x}+3e^{-4y}+C'=0$
Final answer: $4e^{3x}+3e^{-4y}+C=0$