Question

Identify the singular point. Find the general solution that is valid for values of \(t\) on either side of the singular point. $$ t^{2} y^{\prime \prime}+3 t y^{\prime}+17 y=0 $$

Short answer

Answer: The general solution for the given differential equation is \(y(t) = a_0\), where \(a_0\) is an arbitrary constant.

Step-by-step solution

Identify the singular point

A singular point for a differential equation is a value where any of the coefficients are not analytic, or where the coefficients have a pole (infinity). In this case, we have the coefficients: $$t^2, \quad 3t, \quad 17$$ All these coefficients are analytic functions everywhere, so there is no singular point. Therefore, there is no need to worry about finding solutions on either side of it.

Find a general solution

Since there is no singular point, we can use the standard method of finding a general solution, like trying to find a series solution, which is of the form: $$y(t) = \sum_{n=0}^\infty a_n t^n$$ Taking the first and second derivatives of the series solution, we get: $$y'(t) = \sum_{n=1}^\infty n a_n t^{n-1}$$ $$y''(t) = \sum_{n=2}^\infty n(n-1) a_n t^{n-2}$$ Substituting the series back into the differential equation, we obtain: $$t^2 \sum_{n=2}^\infty n(n-1) a_n t^{n-2} + 3t \sum_{n=1}^\infty n a_n t^{n-1} + 17 \sum_{n=0}^\infty a_n t^n = 0$$ Now, we can simplify this expression by eliminating the sums: $$\sum_{n=2}^\infty n(n-1) a_n t^n + \sum_{n=1}^\infty 3n a_n t^n + \sum_{n=0}^\infty 17a_n t^n = 0$$ Since the sum is zero, the coefficients must be zero. We can then set up a recurrence relation: $$n(n-1) a_n + 3n a_n + 17a_n = 0$$ Simplifying this, we have: $$(n^2 - n +3n + 17) a_n = 0$$ $$\Rightarrow (n^2 + 16) a_n = 0$$ Since \(n^2 + 16\) is never zero, this means that all the coefficients \(a_n\) must be zero. Thus, the series solution is: $$y(t) = \sum_{n=0}^\infty a_n t^n = a_0$$ So, the general solution is a constant function, with the constant \(a_0\) as an arbitrary constant: $$y(t) = a_0$$

Key concepts

Singular Points in Differential Equations

In the realm of second order differential equations, a singular point is a point where the coefficients of the terms in the equation become non-analytic. This could mean that they have a discontinuity or a pole (where they approach infinity). It is crucial to identify singular points because they often indicate boundaries where the behavior of the solution changes.

Let's consider the differential equation: \[ t^{2} y'' + 3t y' + 17y = 0 \]Here, the coefficients are \( t^2 \), \( 3t \), and \( 17 \).

• All these coefficients are analytic for all real values of \( t \), meaning they are smooth and continuous.
• Thus, there is no point where the equation 'breaks down'.

There are no singular points in this case.

Knowing about singular points helps us determine if there is a need to find solutions separately in different regions. Here, since there are none, we can seek solutions valid for all \( t \).

Series Solution Method

The series solution is a powerful method for solving differential equations, especially when dealing with complex problems that can't be solved using simple algebraic manipulations.

This involves expressing the solution \( y(t) \) as an infinite sum of terms:\[ y(t) = \sum_{n=0}^\infty a_n t^n \]

• We find the derivatives \( y'(t) \) and \( y''(t) \) in terms of this series.
• We then substitute these back into the original differential equation.

This allows us to create a simpler equation by combining like terms from each power of \( t \).

Finally, we look for a pattern or a recurrence relation by setting the sum of coefficients to zero, as each coefficient represents a balance of terms for each power.

In our case, this method led us to a clean result indicating that the series terms beyond a constant are zero, showcasing the solution's simplicity.

Finding the General Solution

Every differential equation has a general solution that covers all possible particular solutions. The general solution includes arbitrary constants, which can be adjusted for specific initial conditions.

In the problem at hand, after confirming no singular points exist, we proceeded with the series solution method.

• Upon simplification, we derived the condition:
• \((n^2 + 16) a_n = 0\) - meaning unless \(n^2 + 16 = 0\), every \(a_n\) must be zero.

This is challenging because \(n^2 + 16\) never equals zero (as \(n^2\) can't be negative), implying all non-zero coefficients beyond \(a_0\) vanish.

Thus, the general solution simplifies to: \[ y(t) = a_0 \]and the function is just a constant. The constant \(a_0\) remains an arbitrary value, representing the freedom in the initial conditions or the specific occurrence of this equation. This highlights how, even in seemingly complex situations, solutions can sometimes be quite straightforward.