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: Understanding the Concept:
First remove the logarithm to get \(dy/dx\) explicitly, then separate variables since the right side splits into a pure function of \(x\) times a pure function of \(y\).

Step 2: Removing the log:
\(\dfrac{dy}{dx}=e^{3x+4y}=e^{3x}\cdot e^{4y}\).

Step 3: Separating variables:
\(e^{-4y}\,dy=e^{3x}\,dx\).

Step 4: Integrating both sides:
\(\displaystyle\int e^{-4y}\,dy=\int e^{3x}\,dx\ \Rightarrow\ -\dfrac{1}{4}e^{-4y}=\dfrac{1}{3}e^{3x}+C_1\).

Step 5: Clearing fractions:
Multiply through by \(-12\): \(3e^{-4y}=-4e^{3x}-12C_1\), i.e. \(4e^{3x}+3e^{-4y}=C\) (renaming the constant).

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