Consider the following second-order ODE

𝑑
2
𝑦
𝑑
𝑥
2
=
𝑥
+
𝑦
+
𝑑
𝑦
𝑑
𝑥

Under which of the following conditions will this be considered an Initial Value Problem?

Question

Consider the following second-order ODE

𝑑
2
𝑦
𝑑
𝑥
2
=
𝑥
+
𝑦
+
𝑑
𝑦
𝑑
𝑥

Under which of the following conditions will this be considered an Initial Value Problem?

BlackTom AI · Accepted Answer

The question asks about when a second-order ODE is considered an Initial Value Problem (IVP).

First, recall the typical criteria for an IVP for a second-order equation: you must be given the value of the function and the value of its first derivative at a specific initial point x = x0. In other words, you need two initial conditions of the form y(x0) = y0 and y'(x0) = v0.

Option analysis:
- Since the problem statement provides y(1) = 10 and dy/dx|_{x=1} = 0.1, there are explicit initial values for both y and y' at the same x0 = 1. This aligns exactly with the standard definition of an IVP for a second-order ODE.
- If either initial condition were missing (for example only y(1) = 10 without a specified y'(1), or only y'(1) = 0.1 without y(1)), the problem would not qualify as an IVP in its strict sense, because two independent initial conditions at the same x0 are required to determine a unique solution for a second-order ODE.
- Another potential pitfall is confusing boundary value problems with IVPs. In boundary value problems, conditions are imposed at two different points (for example y(a) and y(b)). Here, both conditions are given at the same point x = 1, which is characteristic of an IVP, not a boundary-value problem.

Given the information in the prompt, the provided initial data satisfies the requirement of supplying both y and y' at a single x0. The reasoning hinges on understanding that a second-order ODE needs two independent initial conditions at the same initial point to specify a unique solution path.

Note on the missing options: The provided data includes an answer string but the option list is empty. In an exam with multiple choices, you would compare each option to the standard IVP criterion described above. The key takeaway is that two initial conditions at x0 are essential, and the given values y(1) = 10 and y'(1) = 0.1 fulfill that requirement, hence constitute an IVP.

BN5206 Week 10 online quiz

Consider the following second-order ODE 𝑑 2 𝑦 𝑑 𝑥 2 = 𝑥 + 𝑦 + 𝑑 𝑦 𝑑 𝑥 Under which of the following conditions will this be considered an Initial Value Problem?

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