Question

Solve the following differential equations by power series and also by an elementary method. Verify that the series solution is the power series expansion of your other solution. $$ y^{\prime}=x y+x $$

Short answer

The general solution is \ y = C e^{x^2/2} \.

Step-by-step solution

- Formulating the Differential Equation

Given the equation \( y' = x y + x \), write it in the standard form for solving by power series.

- Assume a Power Series Solution

Assume that \( y \) can be written as a power series: \[ y = \sum_{n=0}^{\infty} a_n x^n \]

- Compute the Derivative

Compute \( y' \): \[ y' = \sum_{n=0}^{\infty} (n+1) a_{n+1} x^n \]

- Substitute into the Differential Equation

Substitute \( y \) and \( y' \) into the original differential equation: \[ \sum_{n=0}^{\infty} (n+1) a_{n+1} x^n = x \sum_{n=0}^{\infty} a_n x^n + x \]

- Combine Terms

Combine all terms involving powers of \( x \) on both sides of the equation: \[ \sum_{n=0}^{\infty} (n+1) a_{n+1} x^n = \sum_{n=0}^{\infty} a_n x^{n+1} + x \]

- Match Powers of \( x \)

Match the coefficients of like powers of \( x \) to obtain a recursion relation: \[ (n+1) a_{n+1} = a_{n-1} + \delta_{n,0} \]

- Determine Coefficients

Determine the values for \( a_n \): \[ a_1 = \frac{a_0}{1}, a_2 = \frac{a_1}{2} = \frac{a_0}{2}, ... \]

- Find General Solution in Power Series Form

Find the general power series solution for \( y \): \[ y = a_0 e^{x^2 / 2} \]

- Solve Differential Equation by Elementary Method

Now solve the differential equation using separation of variables: \[ \frac{dy}{dx} = xy + x \implies \frac{dy}{dx} - xy = x \implies y' - xy = x \]

- Integrate Both Sides

Integrate each term: \[ y = \int x e^{x^2/2} dx = e^{x^2/2} \int x dx = x e^{x^2/2} + C e^{x^2/2} \]

- Verify Power Series Solution

Verify the power series solution matches the elementary method solution by considering the series expansion: \[ y = e^{x^2/2} \]

Key concepts

Power Series Solutions

A power series solution is a method to solve differential equations. We assume the solution can be written as a power series:
\[ y = \sum_{n=0}^{\infty} a_n x^n \]
This sum represents an infinite series where each term is a power of x with a corresponding coefficient an. We compute the derivative of y:


\[ y' = \sum_{n=0}^{\infty} (n+1) a_{n+1} x^n \]
We then substitute y and y' into the differential equation and combine like terms to get a recursion relation, which helps us determine the series coefficients. In essence, using power series solutions transforms solving differential equations into solving algebraic equations for the series coefficients.

Elementary Methods in Differential Equations

Elementary methods are simple, straightforward techniques used to solve differential equations. One common technique is separation of variables. This method involves rearranging the equation such that all terms involving one variable appear on one side and all terms involving the other variable appear on the opposite side. For example, for the differential equation:
\[ \frac{dy}{dx} = xy + x \]

We rearrange it as:
\[ y' - xy = x \]
The next step involves integrating both sides:
\[ \int y' dx - \int xy dx = \int x dx \]

By solving these integrals, we obtain the solution to the differential equation. Elementary methods are typically easier to understand and apply to basic differential equations, making them a fundamental tool in the study of calculus and differential equations.

Recursive Relations in Power Series

Recursive relations in power series are crucial because they allow us to determine the coefficients of the series. After substituting the power series into the differential equation and matching the powers of x, we achieve a recursive formula: \[ (n+1) a_{n+1} = a_{n-1} + \delta_{n,0} \]
Here, \delta_{n,0} is the Kronecker delta, which is 1 if n = 0 and 0 otherwise. This relation helps us determine each coefficient based on its predecessors:
\[ a_1 =\frac{a_0}{1}, \ a_2 =\frac{a_1}{2} = \frac{a_0}{2}, \ ... \]
Recursive relations thus simplify the process of finding the series solution by breaking it down into simpler steps, allowing us to compute each coefficient using previous ones. These coefficients then build up the complete power series solution.

Integration Techniques

Integration techniques allow us to solve differential equations analytically. One common technique is integration by parts, which is often useful when the integrand is a product of two functions. The formula for integration by parts is: \[ \int u dv = uv - \int v du \]
Another essential method is substitution, where a new variable is introduced to simplify the integral. Suppose we have to integrate:
\[ \int x e^{x^2/2} dx \]
We can use substitution by letting \( u = x^2/2 \), and thus \( du = x dx \). The integral then becomes:
\[ \int e^u du \]
Which simplifies to:
\[ e^u + C = e^{x^2/2} + C \]
Integration techniques, therefore, help us find solutions to differential equations and compute definite and indefinite integrals, which are crucial for determining general solutions.