1. Let f(x) and g(x) be two continuously differentiable functions that satisfy 0≤ f(x) < g(x) for all > 0. Now, consider d x₁(1)= f(x₁(t)) and x2 (1) = g(x2 (1)) dt with x₁ (0) = x₂(0) = 1. Show that x₁ (1) ≤ x₂ (1) for all t > 0. Next, show that the solution of diverges to infinity in finite time Hint: Consider the solution of d =y(t) = 1+y¹⁰, y(0)= 1 dt d 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 > 0. Now, consider d x₁(1)= f(x₁(t)) and x2 (1) = g(x2 (1)) dt with x₁ (0) = x₂(0) = 1. Show that x₁ (1) ≤ x₂ (1) for all t > 0. Next, show that the solution of diverges to infinity in finite time Hint: Consider the solution of d =y(t) = 1+y¹⁰, y(0)= 1 dt d 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
x₁(t)= f(x₁ (t)) and
with x₁ (0)=x₂(0) = 1. Show that x₁ (1) ≤x₂ (1) for all t > 0.
Next, show that the solution of
diverges to infinity in finite time
Hint: Consider the solution of
d
x2 (1) = g(x2 (1))
d
y(t) =1+y¹0, y(0) = 1
x(t) = 1 + x²,
x(0) = 1.
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