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.

e²ˣy' + 2e²ˣ y = 2x

Integral Equations

In Exercises 7–12, write an equivalent first-order differential equation

and initial condition for y.

y = ln x + ∫ₓᵉ √ (t² + (y(t))²) dt

Using Euler’s Method

In Exercises 15–20, use Euler’s method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.

y' = 2xy + 2y, y(0) = 3, dx = 0.2

Solve the homogeneous equations in Exercises 5–10. First put the equation in the form of a homogeneous equation.

(x sin y/x - y cos y/x)dx + (x cos y/x) dy = 0

Solve the homogeneous equations in Exercises 5–10. First put the equation in the form of a homogeneous equation.

(x²+y²)dx + xy dy = 0

Solve the homogeneous equations in Exercises 5–10. First put the equation in the form of a homogeneous equation.

(x.exp(y/x) + y)dx - x dy = 0

Solve the homogeneous equations in Exercises 5–10. First put the equation in the form of a homogeneous equation.

y' = y/x + cos ((y-x)/x)

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

In Exercises 1–22, solve the differential equation.

(x + 3y²) dy + y dx = 0 (Hint: d(xy) = y dx + x dy)

In Exercises 1–22, solve the differential equation.

(1+eˣ) dy + (yeˣ + e⁻ˣ) dx = 0

In Exercises 1–22, solve the differential equation.

y' = sin³ x cos² y

In Exercises 1–22, solve the differential equation.

y' = eʸ/xy

In Exercises 1–22, solve the differential equation.

dy + x(2y - e^(x-x²))dx = 0

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

y dx + (3x - xy + 2)dy = 0, y(2) = -1, y < 0

In Exercises 1–22, solve the differential equation.

y' = xy ln x ln y

In Exercises 1–22, solve the differential equation.

xy' + 2y = 1 - x⁻¹

In Exercises 1–22, solve the differential equation.

y' = (y²-1)x⁻¹

In Exercises 1–22, solve the differential equation.

y' = xeʸ√(x-2)

In Exercises 1–22, solve the differential equation.

y' = xeˣ⁻ʸ csc y

In Exercises 1–22, solve the differential equation.

x dy - (x⁴ - y) dx = 0

In Exercises 43 and 44, let S represent the pounds of salt in a tank at time t minutes. Set up a differential equation representing the given information and the rate at which S changes. Then solve for S and answer the particular questions.

Pure water flows into a tank at the rate of 4 gal/min, and the well-stirred mixture flows out of the tank at the rate of 5 gal/min. The tank initially holds 200 gal of solution containing 50 pounds of salt.

c. When will the tank have exactly 5 pounds of salt and how many gallons of solution will be in the tank?

In Exercises 1–22, solve the differential equation.

sec x dy + x cos² y dx = 0

In Exercises 43 and 44, let S represent the pounds of salt in a tank at time t minutes. Set up a differential equation representing the given information and the rate at which S changes. Then solve for S and answer the particular questions.

Pure water flows into a tank at the rate of 4 gal/min, and the well-stirred mixture flows out of the tank at the rate of 5 gal/min. The tank initially holds 200 gal of solution containing 50 pounds of salt.

b. How many pounds of salt are in the tank after 1 minute? after 30 minutes?

In Exercises 1–22, solve the differential equation.

x dy + (3y - x⁻² cos x) dx = 0, x > 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) .

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

b. as a separable equation.