13. Intro to Differential Equations

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

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

y'(t) = eᵗʸ, y(0) = 1

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

y'(t) = yeᵗ, y(0) = −1

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

y(t) = sec² t/(2y), y(π/4) = 1

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. The general solution of the equation yy'(x) = xe⁻ʸ can be found using integration by parts.