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

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.

(t-1)³ ds/dt + 4(t-1)²s = t+1, t >1

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' = xy ln x ln y

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' = eʸ/xy

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.

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

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.

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.