Question

Solve this Differential Equation: dy/dx = x²/(1 + y²)

A. y = [2e^(1-x) - 1]
B. y = 1 - 2x
C. y = ln(x² - 1)
D. y = x³ + ln(x)
E. No solution

Answer 1

To solve the differential equation dy/dx = x²/(1 + y²), we can separate variables and integrate both sides to obtain y = tan((1/3)x³ + C).

Step-by-step explanation:

To solve the differential equation dy/dx = x²/(1 + y²), we can separate variables and integrate both sides. Here's the step-by-step solution:

1. Multiply both sides by (1 + y²) to get: (1 + y²)dy = x²dx
2. Integrate both sides: ∫(1 + y²)dy = ∫x²dx
3. The integral of (1 + y²)dy is arctan(y), and the integral of x²dx is (1/3)x³. So, we have: arctan(y) = (1/3)x³ + C, where C is the constant of integration.
4. To solve for y, take the inverse tangent (arctan) of both sides: y = tan((1/3)x³ + C)

Therefore, the solution to the differential equation is y = tan((1/3)x³ + C).