(a) Use the method in Exercise 7.4.13 to solve an Euler...

(a) Use the method in Exercise 7.4.13 to solve an Euler equation whose characteristic equation has a double root $r_1=r_2=r$. (b) Solve the specific equations
(i) $x^2 u^{\prime \prime}-x u^{\prime}+u=0$,
(ii) $\frac{d^2 u}{d x^2}+\frac{1}{x} \frac{d u}{d x}=0$.

Key Concepts

Euler-Cauchy (Equidimensional) Equation

This type of differential equation has the form x²u'' + axu' + bu = 0, where the coefficients are powers of the independent variable that match the order of the derivative. The structure of these equations allows them to be transformed into constant-coefficient equations by proposing a solution of the form u(x) = x^r, which leads to an algebraic characteristic equation.

Characteristic Equation

When substituting u(x) = x^r into an Euler-Cauchy equation, the derivatives bring down factors of r that, after simplification, yield a quadratic (or higher order) polynomial in r. The roots of this characteristic equation determine the form of the general solution of the differential equation.

Repeated Roots in Euler Equations

If the characteristic equation for an Euler-Cauchy equation has a double (repeated) root r, the general solution cannot simply be a linear combination of x^r terms. Instead, one solution is x^r, and a second, linearly independent solution is obtained by multiplying by a logarithmic factor, resulting in u(x) = C?x? + C?x? ln(x). This method provides the necessary second solution in cases where standard approaches yield dependent solutions.

Reduction to Separable Equations

In some problems, a higher order differential equation can be reduced to a first order separable equation by an appropriate substitution (such as setting v = u' or rewriting the equation as the derivative of a product). This approach simplifies the integration process and is particularly useful when the differential equation displays a structure that can be expressed as a total derivative.

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