082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

19.3 First-order equations: simple equations and separation of variables 541
Example19.7 Find the general solution of dy
dx = 3cos2x.
Solution Given that dy
dx = 3cos2x, theny = ∫ 3cos2xdx = 3 sin2x +C. This is the required
2
general solution.
If dy
dx =f(x)theny= ∫
f(x)dx.
19.3.2 Separationofvariables
When the function f on the r.h.s. of the equation depends upon both independent and
dependentvariablestheapproachofSection19.3.1isnotpossible.However,first-order
equations which can be writteninthe form
dy
= f(x)g(y) (19.1)
dx
forman important class known asseparable equations. For example,
dy
dx = 3x2 e −4y
isaseparable equation forwhich
f(x) = 3x 2 and g(y) = e −4y
To obtain a solution wefirst divide both sides of Equation (19.1) byg(y) togive
1 dy
g(y) dx =f(x)
Integrating both sides with respecttoxyields
∫ ∫ ∫
1 dy
g(y) dx dx = 1
g(y) dy = f(x)dx
Theequationisthensaidtobeseparated.Ifthelasttwointegralscanbefound,weobtain
arelationshipbetweenyandx,althoughitisnotalwayspossibletowriteyexplicitlyin
termsofxasthe following examples will show.
Separationofvariables:
Thesolution of dy = f (x)g(y) isfound from
dx
∫ ∫
1
g(y) dy = f(x)dx