1. Let f(x) and g(x) be two continuously differentiable functions that satisfy 0≤ f(x) < g(x) for all x > 0. Now, consider d d x₁ (1) = f(x₁ (1)) and x2(t) = g(x2(t)) dt with x₁ (0)=x₂ (0) = 1. Show that x₁ (t) ≤ x₂ (t) for all t > 0. Next, show that the solution of d diverges to infinity in finite time Hint: Consider the solution of y(t)=1+y¹0, y(0)= 1 d dt =x(t)=1+x², x(0) = 1.
1. Let f(x) and g(x) be two continuously differentiable functions that satisfy 0≤ f(x) < g(x) for all x > 0. Now, consider d d x₁ (1) = f(x₁ (1)) and x2(t) = g(x2(t)) dt with x₁ (0)=x₂ (0) = 1. Show that x₁ (t) ≤ x₂ (t) for all t > 0. Next, show that the solution of d diverges to infinity in finite time Hint: Consider the solution of y(t)=1+y¹0, y(0)= 1 d dt =x(t)=1+x², x(0) = 1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:1. Let f(x) and g(x) be two continuously differentiable functions that satisfy
0≤ f(x) < g(x)
for all x > 0. Now, consider
d
d
dt
= x₁(t) = f(x₁(t)) and x2(t) = g(x2(t))
with x₁ (0)=x₂ (0) = 1. Show that x₁ (t) ≤ x₂ (t) for all t > 0.
Next, show that the solution of
d
diverges to infinity in finite time
Hint: Consider the solution of
y(t)=1+y¹0, y(0) = 1
d
dt
=x(t)=1+x², x(0) = 1.
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