Page 146 - DMTH202_BASIC_MATHEMATICS_II

Unit 10: Homogeneous Equations

Notes
dy ax  by c

 A differential equation of the form  can be reduced to the homogeneous

dx Ax  By C
a b
form when  .
A B
dy ax  by c

 A differential equation of the form  can be reduced to the homogeneous
dx Ax  By C

a b
form when  , i.e., aB – bA = 0.
A B
10.4 Keywords

Homogeneous Equation: Homogeneous equation is just an equation where both coefficients of
the differentials dx and dy are homogeneous.
Homogeneous Functions: Homogeneous functions are defined as functions where the sums of
the powers of each term are the same.

10.5 Review Questions

Solve the following differential equations:
2
2
1. (x + y )dx = 2xydy
3
2. x ydy + (x + x y– 2xy – y ) dx = 0
3
2
2
2
3. (1 + e ) dx + e (1 – x/y) dy = 0
x/y
x/y
4. y(8x – 9y) dx + 2x(x – 3y) dy = 0
5. (x – 2xy + 3y ) dx + (y + 6xy – x )dy = 0
2
2
2
2
6. (y dx + x dy) x cos (y/x) = (xdy – ydx) y sin (y/x)
3
2
7. x yd – (x + y ) dy = 0
3
2
2
8. xdy  ydx  x  y dx given that y = 1 when  3x
  y  2 y   2 y 


9.  x tan    y sec   dx  x sec   dy  0
  x   x   x 

 
10. xdx  sin 2 y  ydx  xdy   0
 x 
dy y
11. dx  x  ye  2( / )
x
y
12.   2x y  dx   2x y 3 dy 0.
13.   1x y  dx  2x 2y 3 dy 0.

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