Question

Solve the given differential equation by means of a power series about the given point \(x_{0} .\) Find the recurrence relation; also find the first four terms in each of two linearly independent solutions (unless the series terminates sooner). If possible, find the general term in each solution. \(2 y^{\prime \prime}+x y^{\prime}+3 y=0, \quad x_{0}=0\)

Short answer

The recurrence relation for the coefficients in the power series representations of the linearly independent solutions is given by: \(a_n = -\frac{1}{2n(n-1)}[(na_{n-1} + 3a_{n-2})]\) for \(n\geq2\) where \(a_0 = 0, a_1 = b_1\), and \(a_2 = b_2\) (arbitrary constants).

Step-by-step solution

Power series representation of y, y' and y''.

We will write the power series representation of y, y', and y'' using the summation notation as follows: \(y(x) = \sum_{n=0}^{\infty} a_{n}x^n\) \(y'(x) = \sum_{n=1}^{\infty} na_{n}x^{n-1}\) \(y''(x) = \sum_{n=2}^{\infty} n(n-1)a_{n}x^{n-2}\) 2. Substituting the power series representation into the given differential equation:

Substituting power series into the differential equation.

Now, we substitute the power series representation of y, y', and y'' into the given differential equation: \(2\sum_{n=2}^{\infty} n(n-1)a_{n}x^{n-2} + \sum_{n=1}^{\infty} na_{n}x^{n} + 3\sum_{n=0}^{\infty} a_{n}x^n = 0\) 3. Equating similar powers of x in the equation to form the recurrence relation:

Equating coefficients of similar powers of x.

We equate the coefficients of similar powers of x to form the recurrence relation as follows: For \(x^0\) term: \(3a_0 = 0 \Rightarrow a_0 = 0\) For \(x^1\) term: \(2a_2 + 3a_1 = 0\) For \(x^n (n\geq2)\) terms: \(2n(n-1)a_n + na_{n-1} + 3a_{n-2} = 0\) 4. Solve for the coefficients in the power series:

Solving for coefficients.

In order to find the coefficients \(a_n\), let's rewrite the recurrence relation for \(n\geq2\) as: \(a_n = -\frac{1}{2n(n-1)}[(na_{n-1} + 3a_{n-2})]\) Since \(a_1\) and \(a_2\) are arbitrary constants, we will denote them as \(b_1\) and \(b_2\), respectively. \(a_1 = b_1\) \(a_2 = b_2\) Following the recurrence relation, we find the other coefficients: \(a_3 = -\frac{1}{2(3)(2)}[(3(2)b_2+3b_1)] = -\frac{b_1 + 2b_2}{4}\) \(a_4 = -\frac{1}{2(4)(3)}[(12b_2 + 9b_1)] = -\frac{3b_1+4b_2}{8}\) \(a_5 = -\frac{1}{2(5)(4)}[(20b_2+27b_1)] = -\frac{27b_1 + 10b_2}{40}\) 5. Write the first four terms in two linearly independent solutions.

Writing two linearly independent solutions.

We can now write the first four terms in two linearly independent solutions: \(y_1(x) = a_1x + a_3x^3 + a_4x^4 + \cdots = b_1(x-\frac{x^3}{4}+\frac{3x^4}{8}+\cdots)\) \(y_2(x) = a_2x^2 + a_3x^3 + a_5x^5 + \cdots = b_2(x^2+\frac{x^3}{2}+\frac{5x^5}{16}+\cdots)\) 6. General term (if possible)

General term for each solution, if possible.

While it is not easy to find an explicit general term for each solution in this case, the solutions can be written using the recurrence relation and the arbitrary constants \(b_1\) and \(b_2\).

Key concepts

Differential Equations

Differential equations are mathematical equations that involve derivatives of a function. They describe the relationship between a function and its derivatives. This is a key area in calculus and is fundamental in modeling situations where there are dynamic changes in systems, such as physics, biology, and engineering. A differential equation can be ordinary (ODE) or partial (PDE), depending on the type of derivatives. In this exercise, we are dealing with an ordinary differential equation (ODE) because it involves ordinary derivatives.
Our task is to solve this differential equation using power series. Power series are expansions of functions into infinite sums, which can be really useful in finding exact or approximate solutions to differential equations. In particular, around a given point, which in this problem is at zero ( $x_0 = 0$ ).

Linearly Independent Solutions

Linearly independent solutions refer to a set of solutions of a differential equation that cannot be written as a linear combination of each other. In simpler terms, no solution in the set can be derived by multiplying or adding any of the other solutions.
In solving differential equations using power series, finding linearly independent solutions is crucial. They provide a complete set of solutions for the differential equation. For second-order linear homogeneous differential equations, like the one we are dealing with here, there are typically two linearly independent solutions.

• First solution ( $y_1(x)$ ): Starts with arbitrary constant $b_1$ in its series.
• Second solution ( $y_2(x)$ ): Begins involving the arbitrary constant $b_2$.

These solutions collectively form the general solution of the differential equation.

Recurrence Relation

The recurrence relation is a crucial part of solving a differential equation with a power series. It provides a rule to calculate the coefficients of the series based on previous coefficients. This makes it easier to construct the full series step-by-step.
For the provided problem, the recurrence relation is devised by equating powers of \(x\) from the expanded series, where:

• \(3a_0 = 0\) for \(x^0\) term, which yields \(a_0 = 0\).
• \(2a_2 + 3a_1 = 0\) for \(x^1\) term.
• For terms \(x^n\) where \(n \geq 2\): \(2n(n-1)a_n + na_{n-1} + 3a_{n-2} = 0\).

By organizing these, the relationship between coefficients is established, making it possible to find coefficients like \(a_3, a_4, a_5\), and so on, in terms of initial coefficients \(b_1\) and \(b_2\).

Arbitrary Constants

Arbitrary constants are constants that appear in the general solution of a differential equation. They are 'arbitrary' because they can be any value and are generally determined from initial or boundary conditions, which set specific physical or geometric constraints on the problem being solved.
In our problem, solutions involve constants $b_1$ and $b_2$ . They arise because the initial coefficients $a_1$ and $a_2$ represent these arbitrary constants. Through the process of solving the recurrence relation with a power series, these arbitrary constants make it possible to express multiple specific solutions from a general formula.

• $b_1$: Drives the solution $y_1(x)$.
• $b_2$: Leads to the solution $y_2(x)$.

These constants highlight the flexibility and breadth of solutions available when solving differential equations using power series approaches.