Solve the differential equation in Exercises 9–22.
19. y²(dy/dx) = 3x²y³ - 6x²

[Use of Tech] Analysis of a separable equation Consider the differential equation yy'(t) = ½eᵗ + t and carry out the following analysis.

c. Graph the solutions in part (b) and describe their behavior as t increases. 

In Exercises 23–28, solve the initial value problem.

y dx + (3x - xy + 2)dy = 0, y(2) = -1, y < 0

Solve the initial value problems in Exercises 67–70 for x as a function of t.

(3t⁴ + 4t² + 1) (dx/dt) = 2√3, x(1) = -π√3/4

Solve the differential equation in Exercises 9–22.

13. (dy/dx) = √y cos²√y

Separate the variables of the following differential equation.

$ysi{n}^2\theta \frac{dy}{d\theta }=1$

Find the particular solution that satisfies the given initial condition

$(y^2+y)e^x{y}^{\prime }=y^3+e^x{y}^3;y\left(0\right)=1$

Find the general solution to the differential equation.

$\frac{dy}{dx}=y\sqrt{x}$

Find the particular solution that satisfies the given initial condition $\frac{dy}{dx}=\sin x\bullet \sec y,y\left(\frac{\pi }{2}\right)=\frac{\pi }{4}$.