BackDirection field analysis Consider the first-order initial value problem y'(t)=ay+b,y(0)=A for t≥0 where a, b, and A are real numbers.
a. Explain why y=−b/a is an equilibrium solution and corresponds to a horizontal line in the direction field.
7–16. Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. Assume C, C1, C2 and C3 are arbitrary constants.
y(t) = C₁ sin4t + C₂ cos4t; y''(t) + 16y(t) = 0
Solve the Bernoulli equations in Exercises 29–32.
x²y' + 2xy = y³
21–32. Finding general solutions Find the general solution of each differential equation. Use C,C1,C2... to denote arbitrary constants.
u''(x) = 55x⁹ + 36x⁷ - 21x⁵ + 10x⁻³
State the order of the differential equation and indicate if it is linear or nonlinear.
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).
Find the particular solution to the differential equation given the initial condition .
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
A first-order equation Consider the equation t² y′(t) + 2ty(t) = e⁻ᵗ
a. Show that the left side of the equation can be written as the derivative of a single term.
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. The direction field for the differential equation y′(t)=t+y(t) is plotted in the ty-plane.
Find the general solution to the differential equation .
Consider the differential equation y'(t)+9y(t)=10.
a. How many arbitrary constants appear in the general solution of the differential equation?
17–20. Verifying solutions of initial value problems Verify that the given function y is a solution of the initial value problem that follows it.
y(t) = 8t⁶ - 3; ty'(t) - 6y(t) = 18, y(1) = 5