Question

\(3 x y^{\prime \prime}+(3 x+1) y^{\prime}+y=0\)

Short answer

Solve using the Frobenius series approach to find the coefficients and form the general solution.

Step-by-step solution

Identify the type of differential equation

The given differential equation is a second-order linear homogeneous differential equation of the form:\[3xy^{\'\prime\'\prime} + (3x+1)y^{\prime}+ y = 0\]

Simplify the equation

Divide every term by 3 to simplify the equation:\[xy^{\'\prime\'\prime} + \left( x + \frac{1}{3} \right) y^{\prime} + \frac{y}{3} = 0\]

Identify solution approach

This second-order differential equation can be solved using the method of Frobenius series due to the variable coefficient on the highest derivative term.

Assume a power series solution

Assume a solution of the form: \[y(x) = x^{r}\sum_{n=0}^{\infty} a_{n}x^{n}\] where \( a_0 eq 0 \).

Substitute and equate coefficients

Substitute the power series solution into the original equation and equate coefficients of each power of \( x \) to derive the recurrence relation for the coefficients \( a_n \). This results in a characteristic equation in \( r \).

Solve the characteristic equation

The characteristic equation derived from substituting the series into the differential equation will yield the necessary values for \( r \) which are critical to finding the general solution.

Find the general solution

Combine the solutions for different \( r \) values to form the general solution of the differential equation.

Key concepts

Power Series Solution

In solving differential equations, using a power series can be highly effective. This involves expressing the solution as an infinite series. It’s particularly handy for equations where other methods don't work well.
We assume our solution has the form oindent \( y(x) = \sum_{n=0}^{\infty} a_{n}x^{n+r} \) where each term is a product of a coefficient \(a_n\) and x raised to the power of \(n + r\).
This series form allows us to tackle variable coefficients and other complexities by breaking the problem down into manageable terms.
Steps involve:

• Substitute the series into the differential equation
• Collect and simplify terms
• Derive recurrence relations or characteristic equations

It’s like peeling an onion, unfolding layer by layer, to get to the core solution.

Method of Frobenius

The Method of Frobenius is a refined technique specifically used to find power series solutions. It's particularly useful when the differential equation has a regular singular point.
The method is indicated by the form of the solution we assume: \y(x) = x^{r}\sum_{n=0}^{\infty} a_{n} x^{n} \( y(x) = x^{r}\sum_{n=0}^{\infty} a_{n}x^{n}\).
We look for solutions near a point (often x = 0) where the equation might otherwise behave unpredictably.
In the Frobenius method:

• Identify the point and write the solution in the assumed form
• Plug this form into the equation
• Match terms of like powers of x
• Solve resulting equations to get the coefficients \(a_n\)

This method extends our toolkit, especially for tougher, non-constant coefficient equations.

Characteristic Equation

The characteristic equation is a crucial step in solving differential equations using power series, especially with the Method of Frobenius. This equation helps us find the exponents \(r\) of the terms in the series.
To find the characteristic equation, we:

• Substitute the series form into the differential equation
• Isolate terms involving the lowest power of x, often coming from the leading coefficient
• Form a polynomial equation where solutions give us possible values for \(r\)

These values for \(r\) are essential as they guide the form of our final series solution.
Typically, the characteristic equation looks something like \ Q(r) = 0\ where solving for \(r\) uncovers essential details for our series.

Recurrence Relation

Recurrence relations are pivotal in constructing the full solution of a differential equation when using series. They systematically determine the coefficients \(a_n\) of each term in our solution.
Here’s how we find them:

• Starting with the power series form, substitute it into the differential equation
• Collect and simplify terms
• Generate equations from coefficients of each power of x
• Recurrence relations link \(a_{n+1}\) to earlier coefficients \(a_{n}\)

This step-by-step process means each term in the series depends on previous terms. It carefully builds towards the complete solution.
Understanding and solving these recurrence relations is like assembling a jigsaw puzzle, bringing together all pieces to see the whole picture.