13. Intro to Differential Equations

Find the particular solution to the differential equation $y^{\prime }=2e^t+4t$ given the initial condition $y\left(0\right)=1$.

In Exercises 1–22, solve the differential equation.

2y' - y = xe^(x/2)

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

x dy/dx + 2y = x² + 1, x > 0, y(1) = 1

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

x dy + (y - cos x) dx = 0, y(π/2) = 0

A first-order equation Consider the equation t² y′(t) + 2ty(t) = e⁻ᵗ

a. Show that the left side of the equation can be written as the derivative of a single term.

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

d. The direction field for the differential equation y′(t)=t+y(t) is plotted in the ty-plane.

Find the general solution to the differential equation $\frac{dy}{dx}=-2x+5x^2$.

Consider the differential equation y'(t)+9y(t)=10.

a. How many arbitrary constants appear in the general solution of the differential equation?

17–20. Verifying solutions of initial value problems Verify that the given function y is a solution of the initial value problem that follows it.

y(t) = 8t⁶ - 3; ty'(t) - 6y(t) = 18, y(1) = 5

In Exercises 1–22, solve the differential equation.

(1+eˣ) dy + (yeˣ + e⁻ˣ) dx = 0

In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.

3. y = 1/x ∫(from 1 to x) e^t/t dt, x²y' + xy = e^x

43. Surrounding medium of unknown temperature A pan of warm water (46°C) was put in a refrigerator. Ten minutes later, the water’s temperature was 39°C; 10 min after that, it was 33°C. Use Newton’s Law of Cooling to estimate how cold the refrigerator was.

A predator-prey model Consider the predator-prey model

x′(t) = −4x + 2xy, y′(t) = 5y − xy

c. Find the equilibrium points for the system.

Solving Initial Value Problems

Solve the initial value problems in Exercises 15–20.

θ dy/dθ + y = sin θ, θ > 0, y(π/2) = 1

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.

xdy/dx + y = e ͯ, x > 0