13. Intro to Differential Equations

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 $\frac{dy}{dx}=\sin x\bullet \sec y,y\left(\frac{\pi }{2}\right)=\frac{\pi }{4}$.

What is a separable first-order differential equation?

Explain how to solve a separable differential equation of the form

g(t)y'(y) = h(t)

5–16. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable.

e⁴ᵗy'(t) = 5

5–16. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable.

y'(t) = eʸᐟ²sin t