Solving Initial Value Problems
Solve the initial value problems in Exercises 15–20.


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

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

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

State the order of the differential equation and indicate if it is linear or nonlinear. 

$y^{{}^{\prime \prime \prime }}+3xy=4\sqrt{x}$

State the order of the differential equation and indicate if it is linear or nonlinear. 

$\frac{d^{2}y}{d{x}^2}-\left(\frac{dy}{dx}\right)(1-x)=2$

State the order of the differential equation and indicate if it is linear or nonlinear. 

${\left(y^{\prime \prime }\right)}^2+6e^t{y}^{\prime }=4t$