Question

. Find the derivative of the solution \(\varphi\) of the equation \(\dot{x}=x^{2}+x \sin t\) with respect to the initial condition \(\varphi(0)=a\) for \(a=0\).

Short answer

Based on the given information, we can approximate the derivative of the solution to the differential equation \(\dot{x}=x^{2}+x\sin t\) at the initial condition \(\varphi(0) = a = 0\) using perturbation methods and power series. Considering the small parameter \(\epsilon\), we can determine an approximate solution \(\varphi(t)\) in terms of power series and then differentiate it with respect to the initial condition. Finally, we can evaluate the derivative of the approximate solution at \(a=0\). Although this result may still contain terms with \(\epsilon\), it approximates the desired derivative at the given point.

Step-by-step solution

Find the general solution to the differential equation

To find the general solution for the given differential equation \(\dot{x}=x^{2}+x\sin t\), we can first rewrite it as: $$\frac{dx}{dt} = x^2 + x\sin t$$ This is a first-order, nonlinear ordinary differential equation (ODE). Although it may not have an elementary solution, we will proceed by finding an approximate solution.

Find an approximate solution to the ODE

For certain classes of nonlinear ODEs, we can solve them using perturbation methods or other approximation techniques. In this case, since the nonlinear term \(x^2\) appears with a linear term \(x\sin t\), it seems appropriate to try a perturbative approach. Let's represent the solution as a power series: $$\varphi(t) = \varphi_0 + \epsilon\varphi_1(t) + \epsilon^2\varphi_2(t) + \cdots$$ where \(\epsilon\) is a small parameter that essentially measures how nonlinear the ODE is. In the limit \(\epsilon \to 0\), the ODE becomes linear and more manageable. We can now substitute the series into the ODE and collect terms with the same powers of \(\epsilon\). In doing so, we obtain a sequence of linear ODEs, which can be successively solved.

Find the derivative of the approximate solution with respect to the initial condition

After obtaining the approximate solution \(\varphi(t)\), we can differentiate it with respect to the initial condition \(a\) using the chain rule: $$\frac{d\varphi(t)}{da} = \frac{d\varphi_0}{da} + \epsilon\frac{d\varphi_1(t)}{da} + \epsilon^2\frac{d\varphi_2(t)}{da} + \cdots$$ Since \(\varphi(0) = a\), it follows that \(\varphi_0 = a\) and hence \(\frac{d\varphi_0}{da} = 1\). To find the other derivatives \(\frac{d\varphi_n(t)}{da}\), we may use the chain rule and the sequence of linear ODEs we obtained in Step 2.

Evaluate the derivative of the approximate solution at \(a=0\)

After obtaining the above derivative, we can now evaluate it at the initial condition \(\varphi(0) = a = 0\): $$\frac{d\varphi(t)}{da}\Big|_{a=0} = \frac{d\varphi_0}{da}\Big|_{a=0} + \epsilon\frac{d\varphi_1(t)}{da}\Big|_{a=0} + \epsilon^2\frac{d\varphi_2(t)}{da}\Big|_{a=0} + \cdots$$ While this result may still contain terms with \(\epsilon\), it is an approximation to the derivative of the solution with respect to the initial condition at the given point. In certain cases, we can determine how accurate the approximation is by analyzing the behavior of the ODE around the given point or comparing the approximate solution to a known exact solution if available.

Key concepts

Nonlinear Differential Equation

A nonlinear differential equation is characterized by the presence of terms that incorporate nonlinear combinations of the unknown function or its derivatives. Unlike linear differential equations, nonlinear ones involve variables to a power other than one or terms that include products or functional compositions of these variables.

For example, the equation \(\dot{x} = x^2 + x \sin t\) mentioned in our exercise is nonlinear due to the term \(x^2\). Such equations are typically more complex and challenging to solve than linear equations because standard methods like superposition do not apply. However, nonlinear differential equations often arise in the modeling of natural phenomena and thus are crucial to understand. When facing a nonlinear ODE, approximate methods, like perturbation techniques, often become invaluable tools for finding solutions.

Perturbation Methods

When exact solutions to nonlinear differential equations are not attainable or are too complex, perturbation methods provide an alternative avenue for obtaining approximate solutions. These methods involve introducing a small parameter, traditionally denoted as \(\epsilon\), which is used to measure the 'degree of nonlinearity' of the problem.

By expanding the solution as a power series in \(\epsilon\), \( \varphi(t) = \varphi_0 + \epsilon\varphi_1(t) + \epsilon^2\varphi_2(t) + \_\cdots \), analysts can often reduce the original complicated ODE into a hierarchy of simpler, linear ODEs that can then be solved iteratively. This powerful technique allows for an understanding of the behavior of the solution in the vicinity of a simpler problem (\(\epsilon = 0\)).

Initial Condition

The initial condition of a differential equation specifies the value of the unknown function (and possibly its derivatives) at a particular starting point. This information is critical for solving an ODE, as it pins down a specific solution from the potentially infinite family of general solutions.

In our exercise, the initial condition is given by \(\varphi(0)=a\), which means that at time \(t=0\), the value of the solution function \(\varphi\) is \(a\). When working with perturbative solutions, the initial conditions are also expanded and incorporated at each order in the series. This step ensures that the solution not only approximately satisfies the differential equation but also aligns with the specified starting values.

Power Series Solution

The power series solution of a differential equation is a way to represent the solution as an infinite sum of terms with increasing powers, which resemble a polynomial but with potentially an infinite number of terms.

In practice, a power series solution involves assuming that the solution can be written as \(\varphi(t) = \sum_{n=0}^\infty \epsilon^n \varphi_n(t)\), with each \(\varphi_n(t)\) being a function that needs to be determined. This series approach is particularly useful when exact solutions are not accessible, as it allows us to approximate the solution as closely as needed by considering enough terms in the series. When dealing with initial conditions, it is often the zeroth term, \(\varphi_0\), that receives the value of the initial condition.

Chain Rule

The chain rule is a fundamental principle in calculus that describes how to take the derivative of a composite function. If a variable \(y\) depends on \(u\) which itself depends on another variable \(x\), then the derivative of \(y\) with respect to \(x\) can be obtained by multiplying the derivative of \(y\) with respect to \(u\) by the derivative of \(u\) with respect to \(x\): \(\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}\).

In the context of differential equations, the chain rule allows us to understand how the solution changes as we vary an initial condition. Thus, when seeking the derivative of \(\varphi(t)\) with respect to the initial condition \(a\), as shown in our exercise, the chain rule serves to differentiate each term in the power series solution with respect to \(a\), giving us insight into the sensitivity of the solution to changes in initial conditions.