Show that the solution of the initial value problem

y' = x + y, y(x₀) = y₀

is

y = -1 -x + (1 + x₀ + y₀) exp(x-x₀).

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.

y' + (tanx)y = cos²x, -π/2 < x < π/2

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.

(1+x)y' + y = √x

Sailing A sailboat is running along a straight course with the wind providing a constant forward force of 50 lb. The only other force acting on the boat is resistance as the boat moves through the water. The resisting force is numerically equal to five times the boat’s speed, and the initial velocity is 1 ft/sec. What is the maximum velocity in feet per second of the boat under this wind?

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

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 1–22, solve the differential equation.

y'/2 + y = e⁻ˣ sin x

In Exercises 1–22, solve the differential equation.

e⁻ˣ dy + (e⁻ˣ y - 4x) dx = 0

In Exercises 1–22, solve the differential equation.

xy' - y = 2x ln x

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?