13. Intro to Differential Equations

17–32. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem.

y'(t) = cos² y, y(1) = π/4

33–38. {Use of Tech} Solutions in implicit form Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one function, be sure to indicate which function corresponds to the solution of the initial value problem.

z(x) = (z² + 4)/(x² + 16), z(4) = 2

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

The scent of a certain air freshener evaporates at a rate proportional to the amount of the air freshener present. Half of the air freshener evaporates within $2$ hours of being sprayed. If the scent of the air freshener is undetectable once $80\%$ has evaporated, how long will the scent of the air freshener last?

$\$2,000$ is invested in an account that earns interest at a rate of $8.5\%$ and is compounded continuously. Find the particular solution that describes the growth of this account in dollars $A$ after $t$ years.  Hint: When interest is compounded continuously, it grows exponentially with a growth constant equivalent to the interest rate.

A pie is removed from an oven and its temperature is $175\text{℃}$ and placed into a refrigerator whose temperature is constantly $3\text{℃}$. After $1$ hour in the refrigerator, the pie is $90\text{℃}$. What is the temperature of the pie $4$ hours after being placed in the refrigerator?

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

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

(t² + 1)³yy'(t) = t(y² + 4)

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

u'(x) = e²ˣ⁻ᵘ