Given the first linearly independent solution y1(x) = x and the point of expansion x0 = 0, find the

second linearly independent solution y2(x) to the equation

x(x + 1)y

00 + xy0 − y = 0

by the method of Frobenius

To find the second linearly independent solution \( y_2(x) \) to the differential equation 

\[ x(x + 1)y'' + xy' - y = 0 \]

given the first solution \( y_1(x) = x \) and the point of expansion \( x_0 = 0 \), we will use the method of Frobenius. The Frobenius method involves looking for solutions in the form of a power series around \( x_0 \). Since the first solution \( y_1(x) \) is already known and the equation is not in its standard form for applying the Frobenius method directly, we will use a technique that involves reduction of order.

Given \( y_1(x) = x \), we assume the second solution has the form:

\[ y_2(x) = v(x) y_1(x) = v(x) x \]

where \( v(x) \) is a function to be determined. 

First, compute the derivatives of \( y_2(x) \):

\[ y_2'(x) = v'(x)x + v(x) \]
\[ y_2''(x) = v''(x)x + 2v'(x) \]

Substitute \( y_2(x) \), \( y_2'(x) \), and \( y_2''(x) \) into the original differential equation:

\[ x(x + 1)(v''(x)x + 2v'(x)) + x(v'(x)x + v(x)) - v(x)x = 0 \]

Simplify each term:

\[ x(x + 1)v''(x)x + 2x(x + 1)v'(x) + x^2 v'(x) + xv(x) - xv(x) = 0 \]
\[ x^2(x + 1)v''(x) + 2x(x + 1)v'(x) + x^2 v'(x) = 0 \]
\[ x^3 v''(x) + x^2 v''(x) + 2x^2 v'(x) + 2x v'(x) + x^2 v'(x) = 0 \]
\[ x^3 v''(x) + x^2 v''(x) + 3x^2 v'(x) + 2x v'(x) = 0 \]

Factor out common terms:

\[ x^2 (x v''(x) + v''(x) + 3 v'(x) + 2 \frac{v'(x)}{x}) = 0 \]
\[ x^2 (x v''(x) + v''(x) + 3 v'(x) + 2 v'(x) \frac{1}{x}) = 0 \]
\[ x^2 (x v''(x) + v''(x) + 5 v'(x)) = 0 \]

Divide by \( x^2 \) (for \( x \neq 0 \)):

\[ x v''(x) + v''(x) + 5 v'(x) = 0 \]
\[ (x + 1) v''(x) + 5 v'(x) = 0 \]

This is a first-order linear differential equation in \( v'(x) \). Let \( u = v'(x) \):

\[ (x + 1) u' + 5 u = 0 \]

This can be solved using an integrating factor. The integrating factor \( \mu(x) \) is:

\[ \mu(x) = e^{\int \frac{5}{x + 1} dx} = e^{5 \ln |x + 1|} = |x + 1|^5 \]

Multiplying through by the integrating factor:

\[ |x + 1|^5 u' + 5 |x + 1|^4 u = 0 \]
\[ \frac{d}{dx} (|x + 1|^5 u) = 0 \]

Integrating both sides with respect to \( x \):

\[ |x + 1|^5 u = C \]
\[ u = \frac{C}{|x + 1|^5} = C (x + 1)^{-5} \]

Thus,

\[ v'(x) = C (x + 1)^{-5} \]

Integrating \( v'(x) \):

\[ v(x) = \int C (x + 1)^{-5} dx = C \int (x + 1)^{-5} dx \]
\[ v(x) = \frac{C}{-4} (x + 1)^{-4} + D \]

Therefore,

\[ y_2(x) = v(x) x = \left( \frac{C}{-4} (x + 1)^{-4} + D \right) x \]

Choosing a convenient constant for simplicity, let \( C = -4 \) and \( D = 0 \):

\[ y_2(x) = \left( \frac{-4}{-4} (x + 1)^{-4} \right) x = x (x + 1)^{-4} \]

Thus, the second linearly independent solution is:

\[ y_2(x) = x (x + 1)^{-4} \]

This gives us the second solution to the differential equation using the method of Frobenius with reduction of order.

Would you like details or have any questions?

Here are five more questions you might want to ask:
1. How do you solve a second-order linear homogeneous differential equation using the characteristic equation?
2. What are the steps to solve a non-homogeneous differential equation using the method of undetermined coefficients?
3. Can you explain the method of variation of parameters for solving differential equations?
4. How do you find the Laplace transform of a given function?
5. What is the process to solve a system of linear differential equations using matrix methods?

**Tip:** When solving differential equations, always check if the given function is a solution by substituting it back into the original equation. This helps to verify the correctness of your solution.

Learn how to find the second linearly independent solution to a differential equation using the method of Frobenius with reduction of order. This detailed solution involves step-by-step computations and application of the Frobenius method. Suitable for advanced undergraduate students and anyone interested in differential equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation