Question

The solution of \( rdx + (x - r^2) dr = 0 \) is

• A. \( r^2x = \frac{r^3}{3} + c \)
• B. \( rx = \frac{r^2}{2} + c \)
• C. \( x = \frac{r^3}{3} + c \)
• D. \( rx = \frac{r^3}{3} + c \)

Hint:

When solving differential equations, always try to separate variables and integrate both sides.

Answer 1

The Correct Option is D

Step 1: Separate the variables.
Rewrite the given equation as: \[ r \, dx + (x - r^2) \, dr = 0 \] Separate the variables \( x \) and \( r \): \[ r \, dx = (r^2 - x) \, dr \]
Step 2: Integrate both sides.
After solving the integral, we find: \[ rx = \frac{r^3}{3} + c \]
Step 3: Conclusion.
Thus, the solution is \( rx = \frac{r^3}{3} + c \), corresponding to option (D).