Question

let \(\phi_{0}(t)=0\) and use the method of successive approximations to solve the given initial value problem. (a) Determine \(\phi_{n}(t)\) for an arbitrary value of \(n .\) (b) Plot \(\phi_{n}(t)\) for \(n=1, \ldots, 4\). Observe whether the iterates appear to be converging. $$ y^{\prime}=t^{2} y-t, \quad y(0)=0 $$

Short answer

Short Answer: The method of successive approximations enables us to find the approximations of the IVP's solution. We start with an initial approximation, substitute it into the given derivative equation, integrate the result, and apply the initial condition y(0)=0 to find the next approximation. The general form of the approximations is given by φ'_n(t) = t^2φ_(n-1)(t) - t. To observe convergence, we can plot the first four approximations, φ_1(t), φ_2(t), φ_3(t), and φ_4(t), and determine if they are getting closer to each other as n increases.

Step-by-step solution

Understand the method of successive approximations

The method of successive approximations involves iteratively updating our approximation of the solution to the IVP. The first approximation is given as \(\phi_0(t)=0\). To find the second approximation \(\phi_1(t)\), we will substitute \(\phi_0(t)\) into the given derivative equation \(y'=t^2y-t\), and integrate the resulting expression with respect to \(t\).

Substitute \(\phi_0(t)\) into the derivative equation

We have \(\phi_0(t)=0\), so we substitute this value of \(y\) into the equation \(y'=t^2y-t\) to get the following: $$ \phi_1'(t)=t^2(0)-t=-t $$

Integrate the first approximation's derivative

Now, we integrate the above expression to obtain \(\phi_1(t)\): $$ \int \phi_1'(t) dt = \int -t dt $$ $$ \Rightarrow \phi_1(t) = -\frac{1}{2}t^2+C_1 $$ We will use the initial condition \(y(0)=0\) to determine the constant \(C_1\).

Apply initial condition to determine \(C_1\)

By using the initial condition \(y(0)=0\), we find the constant \(C_1\): $$ \phi_1(0) = -\frac{1}{2}(0)^2+C_1=0 $$ $$ \Rightarrow C_1=0 $$ Thus, our first approximation becomes \(\phi_1(t)=-\frac{1}{2}t^2\).

Find the general form of \(\phi_n(t)\)

We can now use the same process to find the general form of \(\phi_n(t)\). We will substitute \(\phi_{n-1}(t)\) into the equation \(y' = t^2 y -t\) and integrate the resulting expression with respect to \(t\). This process is iterative and will be repeated for all approximations. $$ \phi_n'(t) = t^2\phi_{n-1}(t) - t $$ After each iteration, we will apply the initial condition \(y(0)=0\) to find the corresponding constant.

Plotting \(\phi_n(t)\) for \(n=1,\ldots, 4\)

To visualize the iteration process and observe the convergence, we will plot the first four approximations \(\phi_n(t)\) by calculating each \(\phi_n(t)\) given the previous approximation. Observe convergence by seeing if the approximations are getting closer to each other as \(n\) increases. (a) Determine \(\phi_n(t)\) for an arbitrary value of \(n\) The iteratively generated approximations for \(\phi_n(t)\) is given by the formula: $$ \phi_n'(t) = t^2\phi_{n-1}(t) - t $$ (b) Plot \(\phi_n(t)\) for \(n=1, \ldots, 4\) For each subsequent approximation, calculate \(\phi_n(t)\) by substituting the previous approximation \(\phi_{n-1}(t)\) into the derivative equation, integrate the resulting expression with respect to \(t\) and apply the initial condition \(y(0)=0\). Plot the four approximations and observe their convergence.

Key concepts

Initial Value Problem (IVP)

An Initial Value Problem (IVP), in the realm of differential equations, is a mathematical query that seeks a function satisfying not only a differential equation but also specific conditions imposed at the start, known as initial conditions. The problem presents itself as a quest to find a unique curve that passes through a given point, governed by an equation relating the rates of change across various variables.

An exemplary IVP would be finding a function, typically denoted by y(t), that solves a differential equation like \(y' = f(t, y)\) while also adhering to an established starting point, such as \(y(t_0) = y_0\). The task is akin to charting a path, already determined at its origin, that faithfully follows a set of directional rules dictated by the differential equation. In the context of successive approximations, the solution to an IVP unfolds through iterative refinements, originating from an initial guess and pursuing closer alignment with the governing rules and initial condition with each iteration.

Differential Equations

Differential equations stand as a cornerstone in the edifice of mathematics, the sciences, and engineering. Defined by an equation embodying derivatives—symbols of change—these mathematical constructs encapsulate the dynamic patterns describing how physical quantities evolve over time or space.

A differential equation such as \(y' = t^2y - t\) succinctly captures an entire family of possible functions y(t), each displaying particular behavioral patterns dependent on time t. In essence, each solution to this equation is a function whose rate of change at any point reveals a specific relationship between the values of the function itself and the independent variable at that point. Simplistically, a differential equation is akin to a recipe: it prescribes a method to prepare a curve that depicts the relationship between variables, where the derivatives guide the curve's shape and behavior.

Integration Techniques

Integration techniques are the mathematical tools we employ to weave together the moment-to-moment changes of a function, as articulated by its derivative, into a comprehensive picture of the function's overall behavior. It's a reverse-engineering purview, taking the fragments of differential change and conceiving the curves they compose.

Within the framework of IVPs, each approximation to our target solution involves integrating a function like \(-t\) to obtain \(\frac{1}{2}t^2\), accompanied by an integration constant that's pinned down using initial conditions. Mastering integration is key to solving differential equations, offering the lens to view the panoramic whole from infinitesimal parts. It's a toolkit for constructing—from the gradient's sketch—the function's portrait, the continuous saga spelled out by varying rates of change.

Convergence of Iterative Methods

The convergence of iterative methods is the assurance that our sequence of successive approximations, each crafted with care, is indeed inching towards the true solution of the IVP. It’s the mathematician’s peace of mind: a theoretical affirmation that our tireless iterations are not a Sisyphean ordeal, but a journey with a destination.

By plotting iterations like \(\frac{1}{2}t^2\) from \(n=1\) to \(n=4\), we are not merely sketching curves; we're witnesses to the dance of convergence, a choreography where each successive approximation, with bated breath, approaches the choregraph's blueprint. It’s pivotal to verify that with each round of iteration, the approximations narrow in on a single function, homing in like determined arrows, ultimately delivering the true essence of the function described by the differential equation.