Show that if (a-1)^2=4 b, then u2=(lnt) t^(1-a) / 2 is a solution of (8).

Name: Problem 7, Chapter 6: Further Mathematics for Economic Analysis
Uploaded: 2026-08-08T02:42:53-08:00
Duration: 2 min 23 s
Channel: Nick Johnson
Description: Show that if (a-1)^2=4 b, then u2=(lnt) t^(1-a) / 2 is a solution of (8).

step 1 All right, so assuming here that equation 8 is t squared u double prime plus a t u prime plus b

Show that if (a-1)^2=4 b, then u2=(lnt) t^(1-a) / 2 is a...

Key Concepts

Logarithmic Solutions and Reduction of Order

When the indicial equation has a repeated root, the second solution is typically constructed using a logarithmic factor multiplying the standard solution. A common result is that if y1 = t^r is one solution, then y2 = t^r ln(t) serves as a second linearly independent solution. This approach is backed by the method of reduction of order, which adjusts for the failure of the straightforward power-function method in the repeated root scenario.

Repeated Roots

In cases where the characteristic equation yields a double (repeated) root, the usual method of forming two linearly independent power-function solutions fails because both would be of the same form. The condition for a repeated root, often derived by setting the discriminant of the quadratic indicial equation to zero, necessitates a different approach to finding a second independent solution.

Euler-Cauchy Differential Equations

These are a class of linear differential equations with variable coefficients that are homogeneous and have the form t^2 y'' + p t y' + q y = 0. They are characterized by the property that a change of variables (often using power functions like t^r) turns the equation into an algebraic equation in r, making them manageable even though the coefficients vary with t.

Indicial (Characteristic) Equation

When substituting a trial solution of the form y = t^r into an Euler-Cauchy differential equation, one obtains an algebraic equation in r known as the indicial or characteristic equation. This equation determines the exponents for the power-function solutions and plays a crucial role in understanding the behavior of the solutions.

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