Find the general solution to the differential equation $\frac{dy}{dx}=-2x+5x^2$.

• A. $y=-x^2+\frac{5}{3}{x}^3+C$
• B. $y=-x^2+10x+C$
• C. $y=-x+5x+C$
• D. $y=-x^{}+\frac{5}{3}{x}^2+C$

Answer: (A)

Verified step by step guidance

1. Step 1: Recognize that the given equation \( \frac{dy}{dx} = -2x + 5x^2 \) is a first-order ordinary differential equation. To solve it, we will integrate both sides with respect to \( x \).
2. Step 2: Write the equation in integral form: \( y = \int (-2x + 5x^2) \, dx + C \), where \( C \) is the constant of integration.
3. Step 3: Break the integral into two separate terms: \( y = \int -2x \, dx + \int 5x^2 \, dx + C \).
4. Step 4: Solve each integral term-by-term: \( \int -2x \, dx = -x^2 \) and \( \int 5x^2 \, dx = \frac{5}{3}x^3 \).
5. Step 5: Combine the results to write the general solution: \( y = -x^2 + \frac{5}{3}x^3 + C \), where \( C \) is the constant of integration.