To solve the differential equation \(\frac{dy}{dx} = 5x^3\), we can integrate both sides with respect to \(x\).
\[ \int dy = \int 5x^3 , dx \]
The left side integrates to:
\[ y = \int 1 , dy = y \]
Now we integrate the right side:
\[ \int 5x^3 , dx = 5 \cdot \frac{x^4}{4} + C = \frac{5}{4}x^4 + C \]
where \(C\) is the constant of integration.
Thus, the general solution to the differential equation is:
\[ y = \frac{5}{4}x^4 + C \]
where \(C\) is an arbitrary constant.