Solve the following differential equations by series and also by an elementary method and verify that your solutions agree. Note that the goal of these problems is not to get the answer (that's easy by computer or by hand) but to become familiar with the method of series solutions which we will be using later. Check your results by computer. $$x y^{\prime}=x y+y$$

A series solution is a method to solve differential equations by expressing the solution as an infinite sum of terms. Assume the solution is in the form of a power series:

• \( y = \sum_{n=0}^{\infty} a_n x^n \)
• Here, \(a_n\) are coefficients that we need to determine.

First, derive the series for \(y'\):

• \( y' = \sum_{n=1}^{\infty} n a_n x^{n-1} \)

Insert these into the differential equation:

• \( x \sum_{n=1}^{\infty} n a_n x^{n-1} = x \sum_{n=0}^{\infty} a_n x^n + \sum_{n=0}^{\infty} a_n x^n\).

Merge the series on both sides and simplify. Now, solve for the \(a_n\) coefficients by comparing terms for each power of \(x\). This process ensures you get an infinite series that satisfies the differential equation.

Elementary methods refer to straightforward techniques for solving differential equations. One such technique is separation of variables. Another is recognizing the structure of a first-order linear differential equation and using its standard solution method. For our equation, start by rewriting it to isolate the derivative:

• \(y' = y + \frac{y}{x}\)

Recognize it as a linear first-order DE. Use separation of variables by dividing both sides:

• \( \frac{dy}{dx} = y(1 + \frac{1}{x})\)

Rewrite and integrate both sides:

• \( \frac{1}{y} dy = (1 + \frac{1}{x}) dx \)

Integrate to find an implicit solution for \(y\). Finally, solve explicitly for \(y\) and include a constant of integration.

A first-order linear differential equation has the form \( y' + P(x)y = Q(x) \). Our problem, after dividing by \(x\), becomes:

• \( y' = y + \frac{y}{x} \)

Rewrite it to fit the standard form:

• \( y' - y(1 + \frac{1}{x}) = 0 \)

Use the integrating factor method to solve this type of DE. Find the integrating factor \( \mu(x) \) which is \( e^{\int P(x)dx} \). Multiply through the equation by \( \mu(x) \) and rewrite it to see that the left-hand side is the derivative of \( \mu(x)y \).Integrate both sides to solve for \(y\). This method is reliable for any first-order linear differential equation, providing a systematic way to find the solution.

Separation of variables involves manipulating the differential equation to isolate functions of \(x\) and \(y\) on opposite sides of the equation. This technique is straightforward if applicable.In our example, the equation simplifies to:

• \( y' = y(1 + \frac{1}{x}) \)

Separate by dividing both sides by \( y \) and multiplying both sides by \(dx\):

• \( \frac{1}{y} dy = (1 + \frac{1}{x}) dx \)

Integrate both sides to find the solution. The process typically yields a natural logarithm on one side and terms involving \(x\) on the other. Rearrange and exponentiate as needed to solve for \( y \). This method is especially useful because it breaks down the solution process into simpler integrations. Verify any constants from integration steps to ensure the solution fits any initial conditions given.