Question

Solve the given initial value problem. What is the interval of existence of the solution? Consider the third order equation \(t^{3} y^{\prime \prime \prime}+\alpha t^{2} y^{\prime \prime}+\beta t y^{\prime}+\gamma y=0, t>0\). Make the change of independent variable \(t=e^{z}\) and let \(Y(z)=y\left(e^{z}\right)\). Derive the third order constant coefficient equation satisfied by \(Y\).

Short answer

Question: Determine the interval of existence of the solution for the linear third-order ODE given by \(t^3y'''+\alpha t^2y''+\beta ty'+\gamma y=0\) after the change of independent variable \(t=e^z\). Answer: The interval of existence for the solution after the change of independent variable is all real numbers \((-\infty, +\infty)\).

Step-by-step solution

Change of Independent Variable

Rewrite \(t\) as \(t=e^z\). Then re-write \(y(t)\) as \(Y(z)=y(e^z)\), so that it is now a function of \(z\) instead of \(t\).

Compute Derivatives

Find the first three derivatives of \(Y\) with respect to \(z\): First derivative, \(Y'(z)\): Using the chain rule, we have \(Y'(z) = \frac{dy}{dz} = \frac{dy}{dt} \frac{dt}{dz} = y'(t) \cdot e^z\). Second derivative, \(Y''(z)\): Again using the chain rule, \(Y''(z) = \frac{d^2y}{dz^2} = \frac{d}{dz}(y'(t)e^z) = y''(t)e^{2z} + y'(t)e^z\). Third derivative, \(Y'''(z)\): One more time using the chain rule, \(Y'''(z) = \frac{d^3y}{dz^3} = \frac{d^2}{dz^2}(y''(t)e^{2z} + y'(t)e^z) = y'''(t)e^{3z} + 3y''(t)e^{2z} + 2y'(t)e^z\).

Substitute and Simplify

Substitute the derivatives \(Y'(z)\), \(Y''(z)\), and \(Y'''(z)\) into the given equation \(t^3y'''+\alpha t^2y''+\beta ty'+\gamma y=0\): \(e^{3z}(e^{3z}y'''+\alpha e^{2z}y''+\beta e^z y' + \gamma y) = 0\) \(e^{6z}y''' + \alpha e^{5z}y''+\beta e^{4z}y' + \gamma e^{3z}y = 0\) Now, divide through by \(e^{3z}\) to get the third order constant coefficient equation: \(e^{3z}y''' + \alpha e^{2z}y'' + \beta e^z y' + \gamma y = 0\)

Interval of Existence

Since the equation we derived does not contain singular points, the interval of existence for the solution to the problem is all \(z\) for which \(t>0\). Since \(t=e^z\), we can solve for the interval in terms of \(z\): \(t=e^z>0 \Rightarrow e^z>0\) Since the exponential function is always positive, the interval of existence for the solution is all real numbers \((-\infty, +\infty)\).

Key concepts

Change of Variables

When solving complex differential equations, especially higher order ones, employing a strategic change of variables can greatly simplify the problem. Here, the variable transformation from \(t\) to \(z\) is done by setting \(t = e^z\) and defining \(Y(z) = y(e^z)\). This alteration essentially shifts the problem to a different domain, often making the equation easier to handle.

• Purpose: It helps in simplifying terms involving products of \(t\) because exponential functions have neat derivative properties.
• Strategy: By converting \(y(t)\) to \(Y(z)\), you work in a domain that can smoothen the calculation and manipulation of derivatives.

By approaching the problem with a new variable, we transform complex terms into more manageable forms. This change focuses on viewing the problem through a new mathematical lens.

Constant Coefficient Differential Equation

A constant coefficient differential equation is an equation where the derivatives of the unknown function appear with constant multipliers. The given third order equation transforms into one when you switch from \(t\) to \(z\).

• Transformation: The change results in derivatives of \(Y(z)\) that are multiplied by powers of \(e^z\), which are then simplified by dividing through the exponential term.
• Result: The resulting equation, \(Y'''(z) + \alpha Y''(z) + \beta Y'(z) + \gamma Y(z) = 0\), has constant coefficients \(\alpha, \beta, \gamma\).

This simplification is crucial since equations with constant coefficients are typically more straightforward to solve. They are easier to analyze and offer well-understood methods for finding solutions.

Initial Value Problem

An initial value problem specifies values of the function and its derivatives at a specific point, providing initial guidance for the problem's solution. In our context, the initial values would be set at a specific \(z\) value. Understanding the interval of existence assures us of where these prescribed initial values effectively determine a unique solution.

• Importance: These initial conditions ensure the solution is correctly tailored to fit the problem from a certain starting point.
• Interval of Existence: With exponential change of variables, \(t = e^z\), the condition \(t > 0\) translates to \(z\) spanning all real numbers.

The interval \((-finity, +\infty)\) confirms that the solution is valid for all real \(z\), ensuring a robust solution over the entire domain without singularities.