Solve the differential equation y^((4)) = x by successive…

Solve the differential equation $y^{(4)} = x$ by successive integration.

Step 1: First integration

\[y''' = \int x\,dx = \frac{x^2}{2} + C_1.\]

Step 2: Second integration

\[y'' = \int\left(\frac{x^2}{2} + C_1\right)dx = \frac{x^3}{6} + C_1 x + C_2.\]

Step 3: Third integration

\[y' = \int\left(\frac{x^3}{6} + C_1 x + C_2\right)dx = \frac{x^4}{24} + \frac{C_1}{2}x^2 + C_2 x + C_3.\]

Step 4: Fourth integration

\[y = \int\left(\frac{x^4}{24} + \frac{C_1}{2}x^2 + C_2 x + C_3\right)dx = \frac{x^5}{120} + \frac{C_1}{6}x^3 + \frac{C_2}{2}x^2 + C_3 x + C_4.\]

Step 5: General solution

Absorbing the numerical factors into the arbitrary constants,

\[y = \frac{x^5}{120} + C_1 x^3 + C_2 x^2 + C_3 x + C_4,\] where $C_1$, $C_2$, $C_3$, and $C_4$ are arbitrary constants.

Step 6: Verification

Differentiating the absorbed form four times: $y' = \frac{x^4}{24} + 3C_1 x^2 + 2C_2 x + C_3$, $y'' = \frac{x^3}{6} + 6C_1 x + 2C_2$, $y''' = \frac{x^2}{2} + 6C_1$, $y^{(4)} = x$.