Problem 29 - Advanced Engineering Mathematics

Question

\(x=0\) is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about \(x=0 .\) Use \((21)\) where necessary and a CAS, if instructed, to find a second solution. Form the general solution on the interval \((0, \infty)\). $$ x y^{\prime \prime}+(1-x) y^{\prime}-y=0 $$

Step-by-Step Solution

Answer

The indicial roots are \( r = 1 \) and \( r = -1 \), differing by an integer. The series solution is: \( y_1(x) = x \sum_{n=0}^{\infty} a_n x^n \).

Step 1: Identify the Differential Equation Type

The given differential equation is a second-order linear differential equation with variable coefficients, specifically identified as a singular differential equation at the point \(x=0\).

Step 2: Write the Differential Equation in Standard Form

Rewrite the equation \(x y'' + (1-x)y' - y = 0\) in the standard form required for series solutions by dividing through by \(x\): \(y'' + \frac{1-x}{x} y' - \frac{1}{x} y = 0\).

Step 3: Assume a Series Solution Form

Since \(x=0\) is a regular singular point, assume a series solution of the form \(y(x) = \sum_{n=0}^{\infty} a_n x^{n+r}\), where \(r\) is to be determined.

Step 4: Substitute the Series Form into the Equation

Substitute the assumed series \(y(x)\) and its derivatives into the differential equation. Compute the derivatives: \(y'(x) = \sum_{n=0}^{\infty} a_n (n+r) x^{n+r-1}\) and \(y''(x) = \sum_{n=0}^{\infty} a_n (n+r)(n+r-1) x^{n+r-2}\).

Step 5: Collect and Simplify Terms

Substitute these derivatives back into the standard form of the differential equation and simplify, grouping terms by powers of \(x^{n+r}\).

Step 6: Determine the Indicial Equation

The indicial equation is found by examining the lowest-power term in the equation, typically from the \(x^{r-2}\) term. Set this to zero: \(r(r-1) + r - 1 = 0\), which simplifies to \(r^2 - 1 = 0\).

Step 7: Solve the Indicial Equation

The solutions to the indicial equation \(r^2 - 1 = 0\) are \(r = 1\) and \(r = -1\). These roots differ by 2, confirming they differ by an integer.

Step 8: Construct the First Series Solution

With \(r = 1\), substitute back into the equation and solve for the coefficients \(a_n\). Assume a leading term \(a_0\), and solve recursively for \(a_n\). This gives one solution as \(y_1(x) = \sum_{n=0}^{\infty} a_n x^{n+1}\).

Step 9: Construct the Second Solution (if needed)

Using a computer algebra system or advanced techniques, find a second, linearly independent solution corresponding to \(r = -1\), potentially involving logarithms.

Step 10: Form the General Solution

Combine the two solutions to form the general solution over the interval \((0, \infty)\): \(y(x) = C_1 y_1(x) + C_2 y_2(x)\), where \(y_1(x)\) and \(y_2(x)\) are the two independent solutions found.

Key Concepts

Regular Singular Point

In differential equations, a singular point is where the equation fails to be analytic, meaning it may not behave calmly in terms of solutions. A "regular singular point" is much like a hiccup in the equation — it disrupts but doesn't prohibit solutions completely. You can still find manageable solutions around these points using specific methods. If we can rewrite the differential equation in a form that highlights this behavior, it is at a regular singular point, meaning that the coefficients of the equation have specific growth more than nominal. This allows for the use of the Method of Frobenius to seek solutions near this point. In the exercise, the regular singular point of interest is at \(x = 0\). By rewriting the differential equation, it was confirmed that such a point exists, making it feasible to explore solutions using series methods.

Indicial Roots

The term "indicial roots" refers to values derived from the indicial equation, which dictate the nature of the series solution around a regular singular point. To uncover these roots, the differential equation is transformed and simplified, eventually leading to an algebraic expression — the indicial equation. Solving this equation provides the indicial roots which are crucial in constructing solutions for differential equations at regular singular points. In our scenario, solving the indicial equation \(r^2 - 1 = 0\) revealed roots \(r = 1\) and \(r = -1\). These roots notably differ by an integer, highlighting their integral role in the behavior of the equation and confirming the Method of Frobenius can lead to a meaningful solution.

Method of Frobenius

The Method of Frobenius is a powerful technique for solving differential equations, especially around regular singular points. It involves assuming a solution in the form of a power series extended by some exponent, specifically \(y(x) = \sum_{n=0}^{\infty} a_n x^{n+r}\), with \(r\) being one of the indicial roots. By substituting this assumed series solution back into the differential equation, you can identify coefficients \(a_n\) in a recursive manner. This method allows us to systematically generate terms of the solution series, yielding a comprehensive series solution around the singular point. In our exercise, with \(r = 1\) substituting back confirmed a valid solution and enabled us to construct a meaningful solution expression around \(x = 0\).

Series Solution

This term refers to the expansion of solutions as infinite series, often used near points where differential equations become tricky, such as regular singular points. By expressing solutions as an infinite sum, each term relates to powers of \(x\) and coefficients generated systematically. For the given differential equation, the series solution is expressed as \(y_1(x) = \sum_{n=0}^{\infty} a_n x^{n+1}\). As the problem specifies, constructing at least one series solution via the Method of Frobenius is key to handling the differential equation at its regular singular point \(x = 0\). The solution becomes especially powerful when used alongside other solutions (or extended using CAS tools for a full picture), forming what becomes the general solution on the interval \((0, \infty)\), covering the behavior broadly across positive values of \(x\).