Question

Consider the initial value problem \(t^{2} y^{\prime \prime}-t y^{\prime}+y=0, y(1)=1, y^{\prime}(1)=1\). (a) What is the largest interval on which Theorem \(3.1\) guarantees the existence of a unique solution? (b) Show by direct substitution that the function \(y(t)=t\) is the unique solution of this initial value problem. What is the interval on which this solution actually exists? (c) Does this example contradict the assertion of Theorem \(3.1\) ? Explain.

Short answer

Based on the step-by-step solution, the largest interval on which Theorem 3.1 guarantees the existence of a unique solution is \(t \in (0, \infty)\). After verifying by direct substitution, we find that \(y(t) = t\) is the unique solution and exists on the interval \(t \in (- \infty, \infty)\). This example does not contradict Theorem 3.1, as the theorem merely guarantees a unique solution in the given interval, but the actual solution can exist on a larger interval.

Step-by-step solution

Understanding Theorem 3.1

Theorem 3.1 guarantees the existence of a unique solution for a given initial value problem if the coefficients of the linear homogeneous differential equation are continuous functions on some interval about \(t_0\). In our given IVP, \(t^2y^{\prime\prime} - ty^\prime + y = 0\), the coefficients of \(y^{\prime\prime}, y^\prime,\) and \(y\) are \(t^2, -t,\) and \(1\) respectively, which are continuous functions for any interval of \(t\) except \(t=0\). Step 2: Find the largest interval where the theorem guarantees a unique solution

Find the largest interval for a unique solution

The initial condition is given as \(y(1)=1\). The largest interval of \(t\) where all coefficients are continuous and include \(t=1\) is \(t \in (0,\infty)\). Step 3: Direct substitution to verify the unique solution

Verify the unique solution

We are given the function \(y(t) = t\). We need to find the first and second derivatives, and then substitute them into our differential equation to check if it is a valid solution. \(y^\prime(t) = 1\) \(y^{\prime\prime}(t) = 0\) Now, substitute \(y(t)\), \(y^\prime(t)\), and \(y^{\prime\prime}(t)\) into the differential equation \(t^2y^{\prime\prime} - ty^\prime + y = 0\): \((t^2)(0) - (t)(1) + (t) = 0\) \(0 - t + t = 0\) \(0 = 0\) This confirms that \(y(t) = t\) is indeed a unique solution to the given initial value problem. Step 4: Determine the interval where the solution exists

Interval of the solution

Since \(y(t) = t\) is a smooth function, it exists for all real values of \(t\). Thus, the interval on which the solution exists is \(t \in (-\infty, \infty)\). Step 5: Evaluate if the example contradicts the theorem

Evaluating the contradiction

Though Theorem 3.1 guarantees the existence of a unique solution in the interval \(t \in (0, \infty)\), our discovered solution \(y(t) = t\) actually exists on an interval that expands beyond what the theorem indicates. It exists on \(t \in (- \infty, \infty)\). However, this does not contradict the assertions of theorem 3.1 as it merely offers a guarantee (or a sufficient condition) for the existence of a unique solution in the given interval. The actual solution could still have a larger interval of existence. The theorem doesn't impose restrictions on the solution interval but rather provides an assurance that if the conditions are met, there will be a unique solution for that interval.

Key concepts

Understanding Differential Equations

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are crucial in modeling the behavior of various physical systems. For instance, they can represent how a population grows over time or how a satellite orbits a planet.

In the initial value problem given, we face a second-order linear homogeneous differential equation, which is a specific type of differential equation. This class of equations can often be solved by finding functions whose derivatives satisfy the equation, given certain initial conditions.

Existence and Uniqueness Theorem

The Existence and Uniqueness Theorem is a fundamental principle in the study of differential equations. It states that under certain conditions, specifically if the coefficients of the differential equation are continuous near the initial point, there will be exactly one solution to an initial value problem.

The theorem provides us with the assurance that our work in finding a solution is not in vain—there is a unique solution, and it can be found within the bounds of the theorem's criteria. The initial value problem provided in the exercise meets the criteria set by the theorem, as it has continuous coefficients near the initial point, hence guaranteeing a unique solution in a certain interval.

Direct Substitution Method

The direct substitution method is a straightforward approach used in verifying whether a proposed function is indeed a solution to a differential equation. By taking the proposed function and its derivatives and plugging them directly into the differential equation, one can evaluate if the equation balances.

In the given example, the function proposed is simple: y(t) = t. After finding the first and second derivatives, substituting back into the original equation showed that it held true, confirming the function as a valid solution to our initial value problem.

Understanding Solution Intervals

The solution interval for a differential equation is the range of the independent variable, in this case, t, over which the solution is defined. The Existence and Uniqueness Theorem provides the largest interval around the initial condition where the solution is guaranteed to be unique. This, however, does not necessarily encompass the full extent of the solution's validity.

As demonstrated in our example, while the theorem indicated a solution would exist uniquely in the interval (0, ∞), the solution y(t) = t actually exists for all real numbers, which is the interval (-∞, ∞). The solution interval can sometimes be larger than what the theorem suggests, as the theorem sets a sufficient condition, not a limiting constraint.