Question

Consider the initial value problem \(\left(1+t^{2}\right) y^{\prime \prime}+y=0, y(0)=1, y^{\prime}(0)=0\). (a) Show, by Theorem \(8.1\), that this initial value problem is guaranteed to have a unique solution of the form \(y(t)=\sum_{n=0}^{\infty} a_{n} t^{n}\), where the series converges absolutely in \(-1

Short answer

In summary, the given initial value problem has a unique solution in the form of a power series for \(-1 < t < 1\). As \(n\) tends to infinity, the ratio of the coefficients of the power series tends to 1, leading to divergence when \(1 < |t|\). Theorem 8.1 and Theorem 3.1 do not contradict each other, as one theorem guarantees the existence of a power series solution in a specific interval while the other guarantees a unique solution for all \(t\) values, without specifying the form of the solution.

Step-by-step solution

Part (a): Showing the existence of a unique solution

We are given the initial value problem \((1+t^2)y'' + y = 0\), with \(y(0)=1\) and \(y'(0)=0\). Let \(y(t) = \sum_{n=0}^{\infty} a_nt^n\), where \(a_0 = 1\) and \(a_1 = 0\). We can find \(y'(t)\) and \(y''(t)\) as follows: \(y'(t) = \sum_{n=1}^{\infty} na_nt^{n-1}\) \(y''(t) = \sum_{n=2}^{\infty} n(n-1)a_nt^{n-2}\) Now substitute these expressions into the initial value problem: \((1+t^{2})\sum_{n=2}^{\infty} n(n-1)a_nt^{n-2} + \sum_{n=0}^{\infty} a_nt^n = 0\) To determine the coefficients \(a_n\), equate the coefficients of the corresponding powers of \(t\). We have: \(a_0 + (2(2-1)a_2) = 0\) \(a_1 + (3(3-1)a_3t + 2a_2)t = 0\) Now, we solve the above equations for \(a_2\) and \(a_3\): \(a_2 = -\frac{a_0}{2} = -\frac{1}{2}\) \(a_3 = -\frac{a_1+2a_2}{6} = 0 \) Thus, the solution of the initial value problem is given by: \(y(t) = a_0 + a_1t + a_2t^2 + a_3t^3+ \cdots = 1 - \frac{1}{2}t^2\) As we see, the solution converges absolutely for \(-1 < t < 1\).

Part (b): Analyzing the divergence of the series solution for \(1 < |t|\)

For the recurrence relation, we have \(r(n)a_{n+2} = s(n)a_n\), where \(r(n)\) and \(s(n)\) are quadratic polynomials. Thus, \(\left|\frac{a_{n+2}}{a_n}\right| = \left|\frac{s(n)}{r(n)}\right|\) As \(n\) tends to infinity, the ratio of the coefficients of \(t^n\) in the sequences \(r(n)\) and \(s(n)\) tends to 1, which implies that the series solution of the initial value problem diverges for \(1 < |t|\).

Part (c): Discussing the contradiction between Theorem 8.1 and Theorem 3.1

Theorem 8.1 states that a unique solution exists in the form of a power series, and Theorem 3.1 states that a unique solution exists for all values of \(t\) when the coefficient functions \(p(t)\) and \(q(t)\) are continuous for \(-\infty < t < \infty\). In our problem, the coefficient functions are continuous for all \(t\), and hence a unique solution should exist according to both theorems. However, the conclusions of these two theorems do not contradict each other. Theorem 8.1 provides a unique solution in the form of a power series that converges absolutely for \(-1 < t < 1\), which means it doesn't guarantee a solution for all values of \(t\). On the other hand, Theorem 3.1 guarantees the existence of a unique solution for all \(t\), but it doesn't specify the form of the solution. In our case, the series solution is not valid for \(1 < |t|\), but there may exist other forms of solutions (not power series) for those values of \(t\). Therefore, both theorems are consistent, and there is no contradiction.

Key concepts

Unique Solution

The concept of a unique solution in the context of differential equations is crucial. It ensures that given specific initial conditions, there is only one function that satisfies both the differential equation and these initial conditions.
For our initial value problem \[(1+t^2)y'' + y = 0, y(0)=1, y'(0)=0\],we look for a unique solution in the form of a power series. This provides assurance that no other solution will satisfy the problem under the given initial conditions. In essence, this uniqueness condition guarantees consistency and predictability in the behavior of the solution across the defined interval.
The existence of a unique solution is often verified using theorems in differential equations, like Theorem 8.1 in our case, which states conditions under which a power series solution is unique and convergent. This approach involves expanding the solution in an infinite series and determining the coefficients that satisfy both the differential equation and the initial values.

Power Series Convergence

Power series convergence is about determining the interval within which a power series represents a function accurately. Specifically, for our problem, the solution is expressed in a power series form: \[y(t) = \sum_{n=0}^{\infty} a_n t^n\].To verify the series converges, we evaluate it within a radius of convergence.
In this exercise, it's shown that the series converges absolutely for \(-1 < t < 1\). This means that if you choose any value of \(t\) within this range, the series will add up to a finite number. Understanding convergence is essential for knowing where the solution is valid and faithfully represents the problem you're trying to solve. It's worth noting that outside this interval, the power series might diverge, rendering it unusable for finding the solution.

Coefficient Functions Continuity

Continuity of coefficient functions plays a significant role in determining the nature of solutions to differential equations. In our scenario, we have coefficient functions such as \(p(t) = 0\) and \(q(t) = (1+t^2)^{-1}\).
These functions are continuous over \(-\infty

Theorems in Differential Equations

Theorems in differential equations provide a framework for understanding when and where solutions exist and their nature. Specifically, Theorem 8.1 and Theorem 3.1 come into play in our problem.

• Theorem 8.1 states conditions for a power series solution's existence, emphasizing convergence within a specific interval.
• Theorem 3.1 extends this by asserting that solutions exist as long as the system's coefficients are continuous. This is irrespective of the form—be it power series or any other.

The perceived contradiction arises from the limitation of power series forms (which may not converge absolutely outside a particular range) against the broader assertion of Theorem 3.1 that solutions, perhaps in different forms, do indeed exist. Thus, multiple perspectives provided by different theorems help us understand all possibilities for solutions across domains.

Theorems in Differential Equations

Theorems in differential equations provide a framework for understanding when and where solutions exist and their nature. Specifically, Theorem 8.1 and Theorem 3.1 come into play in our problem.

• Theorem 8.1 states conditions for a power series solution's existence, emphasizing convergence within a specific interval.
• Theorem 3.1 extends this by asserting that solutions exist as long as the system's coefficients are continuous. This is irrespective of the form—be it power series or any other.

The perceived contradiction arises from the limitation of power series forms (which may not converge absolutely outside a particular range) against the broader assertion of Theorem 3.1 that solutions, perhaps in different forms, do indeed exist. Thus, multiple perspectives provided by different theorems help us understand all possibilities for solutions across domains.