2. Slow moving landslides can move ('creep') at rates ranging from millimetres to several metres per year. Although they rarely claim lives, they can cause major damage to property and infrastructure. Depending on seasonal rainfall, they can accelerate or decelerate. Figure 2: A slow-moving landslide in Ponzano, Italy. Image credit: https://www.geoengineer.org/ne ws/italian-village-is-being-torn-apart-by-slow-moving-landslide In risk mitigation and early warning systems for landslide hazards, an important property of landslides that is monitored using radar technology is the so-called inverse velocity, y(t) = v(t) where V(t) is the landslide's velocity. Consider a slow moving landslide like that in the village of Ponzano, Italy (Figure 2). Suppose the inverse velocity y(t) is governed by the equation 4y -y² +45 dy dt where t is the time in years since the start of monitoring. (a) Write this ODE in form dy dt arctant et/2 (t≥ 0, y > 0) + P(t) y = Q(t) yn for some n, and use a suitable substition u = ya to reduce this differential equation to a linear first order ODE. Hint: look in the lecture slides for a suitable substitution to use for Bernoulli's equation. (b) Solve the ODE to find y(t) in terms of t, given the initial condition y(0) = 2. (c) Is the landslide moving faster or slower at t = 1 than at the start of monitoring? You might need to use a calculator or computer to help with this.
2. Slow moving landslides can move ('creep') at rates ranging from millimetres to several metres per year. Although they rarely claim lives, they can cause major damage to property and infrastructure. Depending on seasonal rainfall, they can accelerate or decelerate. Figure 2: A slow-moving landslide in Ponzano, Italy. Image credit: https://www.geoengineer.org/ne ws/italian-village-is-being-torn-apart-by-slow-moving-landslide In risk mitigation and early warning systems for landslide hazards, an important property of landslides that is monitored using radar technology is the so-called inverse velocity, y(t) = v(t) where V(t) is the landslide's velocity. Consider a slow moving landslide like that in the village of Ponzano, Italy (Figure 2). Suppose the inverse velocity y(t) is governed by the equation 4y -y² +45 dy dt where t is the time in years since the start of monitoring. (a) Write this ODE in form dy dt arctant et/2 (t≥ 0, y > 0) + P(t) y = Q(t) yn for some n, and use a suitable substition u = ya to reduce this differential equation to a linear first order ODE. Hint: look in the lecture slides for a suitable substitution to use for Bernoulli's equation. (b) Solve the ODE to find y(t) in terms of t, given the initial condition y(0) = 2. (c) Is the landslide moving faster or slower at t = 1 than at the start of monitoring? You might need to use a calculator or computer to help with this.
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:2. Slow moving landslides can move ('creep') at rates ranging from millimetres to several metres
per year. Although they rarely claim lives, they can cause major damage to property and
infrastructure. Depending on seasonal rainfall, they can accelerate or decelerate.
Figure 2: A slow-moving landslide in Ponzano, Italy. Image credit: https://www.geoengineer.org/ne
ws/italian-village-is-being-torn-apart-by-slow-moving-landslide
In risk mitigation and early warning systems for landslide hazards, an important property of
landslides that is monitored using radar technology is the so-called inverse velocity, y(t) = v(t)
where V(t) is the landslide's velocity.
Consider a slow moving landslide like that in the village of Ponzano, Italy (Figure 2). Suppose
the inverse velocity y(t) is governed by the equation
4y -y² +45
dy
dt
where t is the time in years since the start of monitoring.
(a) Write this ODE in form
dy
dt
arctant
et/2
(t > 0, y > 0)
+ P(t) y = Q(t) yn
for some n, and use a suitable substition u = ya to reduce this differential equation to a
linear first order ODE.
Hint: look in the lecture slides for a suitable substitution to use for Bernoulli's equation.
(b) Solve the ODE to find y(t) in terms of t, given the initial condition y(0) = 2.
(c) Is the landslide moving faster or slower at t = 1 than at the start of monitoring?
You might need to use a calculator or computer to help with this.
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