978-3-319-17852-3

22 1. First-Order Differential Equations
15. Show how the initial value problem x ′ = f(t, x), x(0) = x 0 , can be transformed
into the integral equation
x(t) =x 0 +
Transform the initial value problem
into an integral equation.
∫ t
0
f(s, x(s))ds.
x ′ =5tx 2 +1, x(1) = 0
16. Using your calculator, plot the function
x(t) =
∫ t
0
e −s2 ds +2, 0 ≤ t ≤ 4.
17. (Leibniz rule) Define the function I(t, b) = ∫ b
a
f(t, s)ds of two variables.
Then
∫
∂I b
∂t = ∂I
f t (t, s)ds, = f(t, b).
∂b
Use the chain rule to show Leibniz rule:
a
∫
d
b(t)
dt I(t, b(t)) = f t (t, s)ds + f(t, b(t))b ′ (t).
a
1.3 Separable Equations
1.3.1 Separation of Variables
A differential equation having the form
x ′ = f(x)g(t), (1.11)
where the right side is a product of a function of x and a function of t, is called
a separable equation.
The procedure to solve (1.11) is simple. To do so we usually write (1.11) in
the notation
dx
dt = f(x)g(t).
We divide by f(x) toobtain
1 dx
f(x) dt = g(t).