Let's solve questions 1, 3, 5, 7, and 9 step-by-step.

Question 1

$\frac{dy}{dx} = -4x + 6x^2$ This is a first-order differential equation. To solve it, we can integrate both sides with respect to $x$.

1. Separate and integrate: $dy = (-4x + 6x^2) \, dx$
2. Integrate: $y = \int (-4x + 6x^2) \, dx$ $y = \int -4x \, dx + \int 6x^2 \, dx$
3. Solve each integral: $y = -4 \cdot \frac{x^2}{2} + 6 \cdot \frac{x^3}{3} + C$ $y = -2x^2 + 2x^3 + C$
4. General Solution: $y = 2x^3 - 2x^2 + C$

Question 3

$4x^3 - 2 \frac{dy}{dx} = 0$ Rearrange to isolate $\frac{dy}{dx}$:

1. Rearrange: $\frac{dy}{dx} = 2x^3$
2. Separate and integrate: $dy = 2x^3 \, dx$
3. Integrate: $y = \int 2x^3 \, dx$ $y = 2 \cdot \frac{x^4}{4} + C$
4. Simplify: $y = \frac{x^4}{2} + C$

Question 5

$y \frac{dy}{dx} = x^2$ Separate variables to solve.

1. Separate variables: $y \, dy = x^2 \, dx$
2. Integrate both sides: $\int y \, dy = \int x^2 \, dx$
3. Solve each integral: $\frac{y^2}{2} = \frac{x^3}{3} + C$
4. Multiply by 2 to simplify: $y^2 = \frac{2x^3}{3} + C$

This is the implicit form of the solution.

Question 7

$\frac{dy}{dx} = 2xy$ This is a separable differential equation.

1. Separate variables: $\frac{1}{y} \, dy = 2x \, dx$
2. Integrate both sides: $\int \frac{1}{y} \, dy = \int 2x \, dx$
3. Solve each integral: $\ln |y| = x^2 + C$
4. Exponentiate both sides to solve for $y$: $y = e^{x^2 + C} = e^C e^{x^2}$
5. Let $e^C = C'$ (a new constant): $y = C' e^{x^2}$

Question 9

$\frac{dy}{dx} = 3x^2 y - 2xy$ Factor out $y$ on the right side:

1. Rewrite: $\frac{dy}{dx} = y (3x^2 - 2x)$
2. Separate variables: $\frac{1}{y} \, dy = (3x^2 - 2x) \, dx$
3. Integrate both sides: $\int \frac{1}{y} \, dy = \int (3x^2 - 2x) \, dx$
4. Solve each integral: $\ln |y| = x^3 - x^2 + C$
5. Exponentiate both sides to solve for $y$: [ y = e^{x^3 - x^2