Euler’s method on more general grids Suppose the solution of the initial value problem y'(t)=f(t,y),y(a)=A is to be approximated on the interval [a, b].
b. Write the first step of Euler’s method to compute u1.

33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.

b. Using the exact solution (also given), find the error in the approximation to y(T) (only at the right endpoint of the time interval).

y′(t) = t/y, y(0) = 4; Δt = 0.1, T = 2; y(t) = √(t² + 16)

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. The general solution of the separable equation y'(t) = t/(y' + 10y⁴) can be expressed explicitly with y in terms of t.

52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.

{Use of Tech} Tumor growth The growth of cancer tumors may be modeled by the Gompertz growth equation. Let M(t) be the mass of a tumor, for t ≥ 0. The relevant initial value problem is:

dM/dt = -rM(t)ln(M(t)/K), M(0) = M₀,

where r and K are positive constants and 0 < M₀ < K.

b. Graph the solution for M₀ = 100 and r = 0.05.

17–20. Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed.

b. In what regions are solutions increasing? Decreasing?

y'(t) = y(y+3)(4-y)

{Use of Tech} Free fall An object in free fall may be modeled by assuming the only forces at work are the gravitational force and air resistance. By Newton’s Second Law of Motion (mass end . acceleration = the sum of external forces), the velocity of the object satisfies the differential equation 

m · v'(t) = mg + f(v)

mass | acceleration | external forces

where f is a function that models the air resistance (assuming the positive direction is downward). One common assumption (often used for motion in air) is that f(v)=−kv^2, for t≥0, where k>0 is a drag coefficient.

c. Find the solution of this separable equation assuming v(0)=0 and 0<v²<g/a. 

Direction 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.

c. Draw a representative direction field in the case that a<0. Show that if A>−b/a, then the solution decreases for t≥0, and that if A<−b/a, then the solution increases for t≥0.