Consider the differential equation .

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

Problem

Consider the differential equation

\frac{dy}{dx} = \frac{y}{x+y}

. (1) Find and simplify an expression for

\frac{d^2y}{dx^2}

in terms of

x

and

y

. (2) Show that the substitution

y = xv

transforms this equation into

x \frac{dv}{dx} = f(v)

, where

f(v)

is a function to be found. (3) Hence find the particular solution of the equation

\frac{dy}{dx} = \frac{y}{x+y}

for which

y = 1

when

x = 1

. Give your answer in the form

x = g(y)