Page 556 - Modelling in Transport Phenomena A Conceptual Approach

536 APPENDIX B. SOLUTIONS OF DIFFERENTIAL EQUATIONS

Multiplication of Eq. (B.1-16) by the integrating factor gives

(B.1-18)

Integration of Q. (B.1-18) gives the solution as
P 'S P
y=- Qpdx+- C (B. 1-19)
where C is an integration constant.

Example B.4 Solve the following differential equation
dY
x- -2y=x3sinx
dx
Solution
The differential equation can be rewritten as
dY 2 2
-- -y = x sinx
dx x
The integrating factor, p, is

Multiplication of Eq. (1) by the integrating factor gives
1 dy
2
--- - = sinz
y
x2 dx 23
Note that Eq. (3) can also be expressed in the form
- Y
d
(-) =sinx
dx x2
Integration of Eq. (4) gives
y= -x2cosx+cx2

B.1.5 Bernoulli Equations
Bernoulli equation has the form

dY
- + P(X) Y = Q(x) yn
dx 72 # 0,1 (B. 1-20)
The transformation
z = yl-n (B. 1-21)
reduces Bernoulli equation to a linear equation, Eq. (B.l-16).