Entry 1.1 – Exact Equations in Differential Equations

QUESTION [21] 2.6.7

Determine whether the equation is exact. If it is exact, find the solution.

ANSWER [Page 576]

SOLUTION

[RP (Research Position)] Before we begin to solve this question, we should acknowledge the condition, \(x>0\). What does this mean? Why is it stated?

[TRUE (Is this true?)]Because the logarithmic function \(f(x)=\log x\) is defined to be only valid for values \(x>0\), the condition is stated.

NOTE \(\log x\equiv\log_e x\equiv\ln x\equiv\frac{\ln x}{\ln e}\). Change of base \(\log_b a=\frac{\ln a}{\ln b}=\frac{\log a}{\log b}\). \(\log x\equiv\ln x\). Base 10 is a TI-84 thing. \(\log x\) and \(\ln x\) are the same. \(\log x\) is base \(e\). For \(\log x\) or \(\ln x\), the logarithmic function is defined to be \(x>0\) for all \(x\). Thus, the acknowledgment of the restriction.

For differential equations, it is important to first cite the theorem(s)/definition(s) prior to attempting to solve.

EXACT EQUATION Theorem 2.6.1

The verbatim theorem above is what we will work from. It is good to recognize that this is the essentially the same exact problem as solving a conservative vector function in multivariable calculus.

Step 1 for Differential Equations: Identify the ODE,