A function is defined implicitly by the differential equation with the initial condition .

MATHEMATICS · Multi-part · CBSE · Mathematics · Differential Equations · Differential Equations

Problem

A function

h

is defined implicitly by the differential equation

\dfrac{dh}{dx}=\dfrac{1}{x}

with the initial condition

h(-2)=3

. (a) Derive a formula for the general solution to the differential equation. (b) Using the initial condition, obtain the particular solution

h(x)

. (c) State the largest open interval containing

x=-2

on which this particular solution is valid, and justify why the solution cannot be extended beyond that interval without modification. (d) Evaluate

\displaystyle \lim_{x\to0^{-}}h(x)

, or explain why the limit does not exist.