Question

Solve the given differential equation by finding an appropriate integrating factor: (2y² - 3x) dx + 2xy dy = 0.

Answer 1

To solve the given differential equation (2y² - 3x) dx + 2xy dy = 0, you can find an integrating factor, which is e^(-x²). Multiply both sides of the equation by the integrating factor, rearrange the terms, integrate both sides, simplify the integral, and solve for y in terms of x.

Step-by-step explanation:

To solve the given differential equation (2y² - 3x) dx + 2xy dy = 0, we need to find an integrating factor. The integrating factor is a function that we multiply the entire equation by in order to make it exact. In this case, the integrating factor is e^(-x²), where e is the base of the natural logarithm.

1. Multiply both sides of the equation by the integrating factor: e^(-x²)(2y² - 3x) dx + e^(-x²)(2xy dy) = 0.
2. Rearrange the terms and notice that the left side is now the derivative of (e^(-x²) y² - e^(-x²)xy): d/dx (e^(-x²) y² - e^(-x²)xy) = 0.
3. Integrate both sides with respect to x: ∫ d/dx (e^(-x²) y² - e^(-x²)xy) dx = ∫ 0 dx.
4. Simplify the integral: e^(-x²) y² - e^(-x²)xy = C, where C is the constant of integration.
5. Finally, solve for y in terms of x: y = (e^(x²) C + x) / e^(x²).