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Chapter 8: Problem 15

\(x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-1\right) y=0\)

Short Answer

Expert verified
The solution to the given differential equation \(x^{2} y^{\prime \prime} + x y^{\prime} + \left( x^{2} -1 \right) y = 0\) is \(y = c_{1}x^{-1} + c_{2}x\) where \(c_{1}\) and \(c_{2}\) are constants.

Step by step solution

01

Identifying the type of differential equation

The given differential equation \(x^{2} y^{\prime \prime} + x y^{\prime} + \left( x^{2} -1 \right) y = 0\) falls under the category of Cauchy-Euler or Equidimensional type. These types of equations can be solved using the substitution \(x = e^{m}\), where \(m\) is a real number.
02

Performing the substitution

Substituting \(x = e^{m}\) into the given equation, the equation becomes \(m^2 y^{\prime \prime} + m y^{\prime} + \left( m^{2} -1 \right) y = 0 \). This simplifies to \(y^{\prime \prime} + y^{\prime} + y = 0\), which is a simple linear homogeneous equation.
03

Solving the linear homogeneous differential equation

This simple linear homogeneous differential equation can be solved by applying the general solution method which yields \(y = c_{1}e^{-m} + c_{2}e^{m}\).
04

Substituting back

We substitute \(m = \ln x\) back into our solution and we get the following solution \(y = c_{1}x^{-1} + c_{2}x\).

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Most popular questions from this chapter

\(x(1-x) y^{\prime \prime}+(1-3 x) y^{\prime}-y=0\)

$$\left(x^{2}-4\right) y^{\prime \prime}+(x+2) y^{\prime}+3 y=0$$

\(f(x)=\frac{1+x}{1-x}, \quad x_{0}=0\)

(a) Construct the Taylor polynomial \(p_{3}(x)\) of degree 3 for the function \(f(x)=1 /(2-x)\) around \(x=0\) $$$$(b) Using the error formula (6), show that $$ \left|f\left(\frac{1}{2}\right)-p_{3}\left(\frac{1}{2}\right)\right|=\left|\frac{2}{3}-p_{3}\left(\frac{1}{2}\right)\right| \leq \frac{2}{3^{5}}$$ $$$$(c) Compare the estimate in part (b) with the actual error $$ \left|\frac{2}{3}-p_{3}\left(\frac{1}{2}\right)\right| $$ $$$$(d) Sketch the graphs of \(1 /(2-x)\) and \(p_{3}(x)\) (on the same axes) for \(-2\)<\(x\)<\(2\)

Classify each singular point (real or complex) of the given equation as regular or irregular. $$ \left(x^{2}+2 x-8\right)^{2} y^{\prime \prime}+(3 x+12) y^{\prime}-x^{2} y=0 $$
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