Question

try to transform the given equation into one with constant coefficients by the method of Problem 34. If this is possible, find the general solution of the given equation. $$ t y^{\prime \prime}+\left(t^{2}-1\right) y^{\prime}+t^{3} y=0, \quad 0

Short answer

If so, find the general solution. Answer: No, it is not possible to transform the given differential equation into a constant coefficient form.

Step-by-step solution

Assuming substitution of Euler-like function

Since we are attempting to simplify the equation, we will use substitution in the form of an Euler-like function: \(y(t)=t^r\). Now let's compute its derivatives then substitute them into the given differential equation. Compute derivatives: - First derivative: \(y'(t) = r t^{r-1}\) - Second derivative: \(y''(t) = r (r-1) t^{r-2}\) Now, substitute these derivatives into the given equation and see if it can be simplified.

Substituting Euler-like function into the given equation

Substitute \(y(t)=t^r\), \(y'(t) = r t^{r-1}\), and \(y''(t) = r (r-1) t^{r-2}\) into the given equation and simplify: \(t(r (r-1) t^{r-2}) + (t^2 -1)(r t^{r-1}) + t^3 (t^r) = 0\) After substituting, we get: \(r(r-1)t^{r} + (t^2-1)r t^r + t^{3+r} = 0\) Now, we simplify further.

Simplify the expression and finding the exponents

All the terms in the equation have common \(t^r\) term, so we factor it out: \(t^r(r(r-1) + (t^2-1)r + t^3) = 0\) Now, since \(t^r \neq 0\) (since \(0<t<\infty\)), we have: \(r(r-1) + (t^2-1)r + t^3 = 0\) This equation has constant coefficients if we can find an \(r\) that makes the last term vanish, i.e., \(t^3 = 0\), but it is not possible as \(t\) is non-zero. Thus, the transformation into an equation with constant coefficients isn't possible.

Key concepts

Euler-like substitution

Euler-like substitution is a technique used to simplify differential equations, particularly those involving variable coefficients. The main idea is to substitute a function in the form of a power, typically using a variable raised to some power, such as

• Example substitution: Choose a function like \( y(t) = t^r \) where \( r \) is an unknown constant.

This substitution helps convert the original equation into a simpler form by allowing direct computation of the derivatives of the assumed function. Once you have the derivatives, they can be substituted back into the original equation.
Substituting an Euler-like form can sometimes transform a complex differential equation into one that is easier to handle, particularly if it becomes a simpler algebraic form.

Differential equation transformation

The process of transforming a differential equation involves seeking ways to convert it into a different form. This method uses substitutions and algebraic manipulations to simplify the problem, potentially revealing solutions that are not obvious at first glance.A common approach is to substitute functions in a manner that lets us use known techniques or solutions of simpler equations. The goal of the original problem was to see if converting it would yield a differential equation with constant coefficients, which would be easier to solve.
For transformations based on Euler-like functions, replacing the original function with one of the form \( y(t) = t^r \) enabled simplifying the equation by collecting like terms and focusing on the powers of \( t \).Although the specific transformation in this problem did not yield a constant coefficient equation, the attempt showcases the idea of restructuring the problem to see if simpler methods can be applied.

Constant coefficients

In differential equations, constant coefficients refer to equations where the coefficients of the derivatives are constant numbers, rather than functions of another variable. This characteristic simplifies the process of finding solutions, as it allows for the application of straightforward methods such as the characteristic equation.

• Simplified Analysis: Constant coefficients mean the form does not change, making it easier to solve.
• Characteristic Roots: You can solve such an equation by finding and solving its characteristic polynomial, making use of exponential functions in the solution.

In the problem at hand, the substitution attempt aimed to work towards achieving this form, thereby converting the given differential equation into a simpler one more amenable to these methods. However, the equation could not be changed to have constant coefficients simply due to the nature of the terms involved.

General solution of differential equations

The general solution of a differential equation provides a complete set of solutions, incorporating any arbitrary constants introduced by the integration of the equation's derivatives. These constants indicate the multitude of solutions that can satisfy the original differential equation's conditions. For linear differential equations with constant coefficients, solving the characteristic equation can yield roots that guide the form of the solution. Depending on the types of roots (real and distinct, real and repeated, or complex), different strategies are used to construct the general solution, such as:

• Real roots: Involves exponentials of the roots multiplied by constants.
• Complex roots: Uses sine and cosine functions to express the solution.
• Repeated roots: Introduces polynomial terms to ensure linearly independent solutions.

While the problem intended to achieve a form with constant coefficients, which was not possible, understanding the general solution involves recognizing that for a given differential equation, there could be multiple valid functions that solve it. This is the versatility offered by finding a "general" rather than a "particular" solution.