Higher Engineering Mathematics

I

Differential equations

48

Linear first order differential equations

48.1 Introduction

An equation of the form

dy

dx

+Py=Q, wherePand

Qare functions ofxonly is called alinear differ-

ential equationsinceyand its derivatives are of the

first degree.

(i) The solution of

dy

dx

+Py=Qis obtained by

multiplying throughout by what is termed an

integrating factor.

(ii) Multiplying

dy

dx

+Py=Qby sayR, a function

ofxonly, gives:

R

dy

dx

+RPy=RQ (1)

(iii) The differential coefficient of a productRyis
obtained using the product rule,

i.e.

d

dx

(Ry)=R

dy

dx

+y

dR

dx

,

which is the same as the left hand side of

equation (1), whenRis chosen such that

RP=

dR

dx

(iv) If

dR

dx

=RP, then separating the variables gives

dR

R

=Pdx.

Integrating both sides gives:

∫

dR

R

=

∫

Pdxi.e. lnR=

∫

Pdx+c

from which,

R=e

∫

Pdx+c=e

∫

Pdxec

i.e.R=Ae

∫

Pdx, whereA=ec=a constant.

(v) SubstitutingR=Ae

∫

Pdxin equation (1) gives:

Ae

∫

Pdx

(

dy

dx

)

+Ae

∫

PdxPy=Ae

∫

PdxQ

i.e. e

∫

Pdx

(

dy

dx

)

+e

∫

PdxPy=e

∫

PdxQ (2)

(vi) The left hand side of equation (2) is

d

dx

(

ye

∫

Pdx

)

which may be checked by differentiating

ye

∫

Pdxwith respect tox, using the product rule.

(vii) From equation (2),

d

dx

(

ye

∫

Pdx

)

=e

∫

PdxQ

Integrating both sides gives:

ye

∫

Pdx=

∫

e

∫

PdxQdx (3)

(viii) e

∫

Pdxis theintegrating factor.

48.2 Procedure to solve differential

equations of the form

dy

dx

+Py=Q

(i) Rearrange the differential equation into the

form

dy

dx

+Py=Q, wherePandQare functions

ofx.

(ii) Determine

∫

Pdx.

(iii) Determine the integrating factor e

∫

Pdx.

(iv) Substitute e

∫

Pdxinto equation (3).

(v) Integrate the right hand side of equation (3)

to give the general solution of the differential