13. Intro to Differential Equations

In Exercises 1–22, solve the differential equation.

dy + x(2y - e^(x-x²))dx = 0

41. Cooling soup Suppose that a cup of soup cooled from 90°C to 60°C after 10 min in a room where the temperature was 20°C. Use Newton’s Law of Cooling to answer the following questions.

a. How much longer would it take the soup to cool to 35°C?

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.

44. Silver cooling in air The temperature of an ingot of silver is 60°C above room temperature right now. Twenty minutes ago, it was 70°C above room temperature. How far above room temperature will the silver be

b. 2 hours from now?

Write the formula for a logistic function that has values between y = 0 and y = 1, crosses the line y = 1/2 at x = 0, and has slope 5 at this point.

In Exercises 1–22, solve the differential equation.

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

In Exercises 1–22, solve the differential equation.

x dy + (3y - x⁻² cos x) dx = 0, x > 0

In Exercises 1–22, solve the differential equation.

x dy - (x⁴ - y) dx = 0

Solve the following initial value problem for u as a function of t:

du/dt + (k/m) u = 0 (k and m positive constants), u(0) = u₀

a. as a first-order linear equation.

28. Derivation of Equation (7) in Example 4

a. Show that the solution of the equation

di /dt + R/Li = V/L

is

i = V/R + Cexp(-(R/L)i) .

Solving Initial Value Problems

Solve the initial value problems in Exercises 15–20.

dy/dt + 2y = 3, y(0) = 1

Solving Initial Value Problems

Solve the initial value problems in Exercises 15–20.

dy/dx + xy = x, y(0) = -6

What integral equation is equivalent to the initial value problem y' = f(x), y(x₀) = y₀?