Question

Find the general solution of the differential equation \(\log\left(\dfrac{dy}{dx}\right)=3x+4y\).

Hint:

Exponentiate to get \(dy/dx=e^{3x}e^{4y}\), separate variables, then integrate.

Answer 1

Step 1: Verifying by implicit differentiation:
Differentiate \(4e^{3x}+3e^{-4y}=C\) w.r.t. \(x\): \(12e^{3x}+3e^{-4y}\cdot(-4)\dfrac{dy}{dx}=0\), i.e. \(12e^{3x}-12e^{-4y}\dfrac{dy}{dx}=0\).

Step 2: Solving for dy/dx:
\(\dfrac{dy}{dx}=\dfrac{12e^{3x}}{12e^{-4y}}=e^{3x}e^{4y}=e^{3x+4y}\).

Step 3: Matching back to the original:
Taking \(\log\) of both sides recovers \(\log\left(\dfrac{dy}{dx}\right)=3x+4y\), the original equation — confirming the solution.

Final Answer:
Verified: \(\boxed{4e^{3x}+3e^{-4y}=C}\).