Question

Find power series solutions in powers of \(x\) of each of the differential equations in Exercises. $$ y^{\prime \prime}+y^{\prime}+3 x^{2} y=0 $$

Short answer

The general power series solution for the given second-order differential equation \(y^{\prime\prime}+y^{\prime}+3x^{2}y=0\) is: $$ y(x)= a_0 \left(\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n} \right) + a_1 \left(\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1} \right) $$ where \(a_0\) and \(a_1\) are the initial coefficients.

Step-by-step solution

Assume a power series solution

First, let's assume a power series solution for the given differential equation: $$ y(x) = \sum_{n=0}^{\infty} a_n x^n $$ where \(a_n\) is the coefficient of the power series.

Find first and second derivatives

Next, we will find the first and second derivatives of the power series solution, denoted by \(y'\) and \(y''\) respectively: $$ y'(x) = \sum_{n=1}^{\infty} n a_n x^{n-1} $$ $$ y''(x) = \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} $$

Plug the power series and its derivatives into the differential equation

Now, we can substitute everything back into the original second-order differential equation: $$ \left(\sum_{n=2}^{\infty} n(n-1) a_n x^{n-2}\right) + \left(\sum_{n=1}^{\infty} n a_n x^{n-1}\right) + 3x^2 \left(\sum_{n=0}^{\infty} a_n x^n\right) = 0 $$ Distributing the terms, we get: $$ \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} + \sum_{n=1}^{\infty} n a_n x^{n-1} + 3 \sum_{n=0}^{\infty} a_n x^{n+2} = 0 $$

Adjust the indices for easier comparison and summation

Let's change the indices in the second and third summations so that the exponents match with the first summation: For the second summation, let \(m=n-1\), then \(n=m+1\), and the second summation becomes: $$ \sum_{m=0}^{\infty} (m+1) a_{m+1} x^m $$ For the third summation, let \(k=n+2\), then \(n=k-2\), and the third summation becomes: $$ 3 \sum_{k=2}^{\infty} a_{k-2} x^k $$ Now rewrite the differential equation with the adjusted indices: $$ \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} + \sum_{m=0}^{\infty} (m+1) a_{m+1} x^m + 3 \sum_{k=2}^{\infty} a_{k-2} x^k = 0 $$

Combine the summations and find a recurrence relation

Combine the summations and equate the corresponding coefficients of \(x^k\): $$ \sum_{n=0}^{\infty} \left((n+2)(n+1) a_{n+2} + (n+1) a_{n+1} + 3 a_n \right) x^n = 0 $$ For this equation to be true, we must have: $$ (n+2)(n+1) a_{n+2} + (n+1) a_{n+1} + 3 a_n = 0 $$ We can rewrite this relation as: $$ a_{n+2} = -\frac{(n+1) a_{n+1} + 3 a_n}{(n+2)(n+1)} $$ This is the recurrence relation for the coefficients of the power series solution.

Express the general power series solution

Using the recurrence relation, we can now express the power series solution in terms of the initial coefficients \(a_0\) and \(a_1\): $$ y(x)= a_0 \left(\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n} \right) + a_1 \left(\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1} \right) $$ This is the general power series solution for the given second-order differential equation.

Key concepts

Differential Equation

A differential equation is a mathematical equation that relates a function with its derivatives. In simpler terms, it's like a recipe where you mix different ingredients — the function and its derivatives — to get a special dish: the equation it satisfies. These equations are powerful tools for understanding how different quantities change over time or space.

In math and science, they help us model physical phenomena like motion, heat, or waves. Differential equations can be ordinary (ODEs) with functions of one variable or partial (PDEs) involving functions of multiple variables. The one we worked with in the exercise, \( y'' + y' + 3x^2 y = 0 \), is an ODE.

The main goal is to find the function \( y(x) \), such that when you fill it into the equation with its derivatives, the equation equals zero — it's like solving a puzzle where you find the missing piece that makes everything fit perfectly.

Recurrence Relation

Recurrence relations are like mathematical instructions that help us find the next term in a series based on previous ones. Think of them as the rules of a game, where knowing the current position allows you to figure out the next move.

In our power series solution, we derived a recurrence relation: \[ a_{n+2} = -\frac{(n+1) a_{n+1} + 3 a_n}{(n+2)(n+1)} \].

This relations tells us how to find each coefficient of our power series by using previously found coefficients, starting from some initial known terms \(a_0\) and \(a_1\).

• Recurrence relations are crucial because they let us generate a sequence without having to solve everything at once.
• They're like the links of a chain — solve one and the rest follows.

With these patterns, we can build whole solutions step by step!

Second-order Differential Equation

Second-order differential equations involve second derivatives, which provide information about the curvature or the way a function accelerates or decelerates. These types of equations play an essential role in describing systems with two layers of change, such as oscillations.

Our equation \( y'' + y' + 3x^2 y = 0 \) is a classic example, with the highest derivative being the second derivative, \( y'' \). Such equations can portray scenarios where phenomena like vibrations or economic changes are modeled by how quickly they speed up or slow down.

Solving second-order ODEs often produces two independent solutions, which can be scaled and added together to create a general solution. This diversity is why these equations are like the Swiss army knives in the toolkit of engineers and scientists.

Series Expansion

Series expansion is a way to express complex functions in terms of simpler polynomial terms, which can be easier to understand or calculate with. It's almost like laying out a detailed blueprint of a complicated structure that shows how it can be built from simple parts.

When solving differential equations with power series, we expand the unknown function, as well as its derivatives, into an infinite sum of terms: \[ y(x) = \sum_{n=0}^{\infty} a_n x^n \].

This method is especially handy when dealing with equations that don't have standard or easy solutions.

The magic of series expansions lies in their flexibility:

• They approximate functions over small ranges, which can be made arbitrarily precise by including more terms.
• They turn complicated behavior into something linear and manageable, especially around points like zero in Taylor series.

By breaking equations down into series expansions, we can tackle them piece by piece, making the complex seem conquerable.