The problem asks to solve the second-order differential equation $y'' = 20x^3$.

1. Problem Description

The problem asks to solve the second-order differential equation $y'' = 20x^3$.

2. Solution Steps

To solve the differential equation $y'' = 20x^3$, we need to integrate twice.

First integration:

$\int y'' \, dx = \int 20x^3 \, dx$

$y' = 20 \cdot \frac{x^4}{4} + C_1$

$y' = 5x^4 + C_1$

Second integration:

$\int y' \, dx = \int (5x^4 + C_1) \, dx$

$y = 5 \cdot \frac{x^5}{5} + C_1x + C_2$

$y = x^5 + C_1x + C_2$

3. Final Answer

The general solution to the differential equation $y'' = 20x^3$ is $y = x^5 + C_1x + C_2$, where $C_1$ and $C_2$ are arbitrary constants.