Page 140 - DMTH202_BASIC_MATHEMATICS_II

Unit 10: Homogeneous Equations

Self Assessment Notes

Fill in the blanks:
1. .................................. is just an equation where both coefficients of the differentials dx and
dy are homogeneous.
2. Homogeneous functions are defined as functions where the .................................. of the
powers of each term are the same.

3. A homogeneous equation can be malformed into a distinguishable equation by a change
of ................................... .

dy f 1  ,x y 
4. An equation of the form  is called a homogeneous function of the
dx f 2  ,x y 
.................................. degree in x and y.

5. If f (x, y) and f (x, y) are homogeneous functions of degree n in x and y, then
1 2
y
y
f 1  ,x y   x n  1   and f 2  ,x y  x n  2  
 
  .................................. .
x
x
   
6. If you contain n variables, and n equations, and the determinant of the system is non-zero,
the it is known as the .................................. solution to them.
7. The function f does not depend on x & y independently but only on their
proportion................................... .
8. If you recognize the fact that an equation is homogeneous you can, in some cases, carry out
a .................................. which will permit you to apply separation of variables to solve the
equation.

9. If we get the solution in terms of v and x replacing v by.................................. we get the
required solution.

State whether the following statements are true or false:
10. The sums of the powers of each term of homogenous function are different.
11. If there are smaller amount of linearly independent equations than there are variables,
then there are other, trivial solutions to the homogeneous equations.

10.2 Equations Reducible to Homogeneous Form

A differential equation of the form

dy ax  by c

 …..(1)
dx Ax  By C

can be reduced to the homogeneous form.

dy ax  by c

Did u know? A differential equation of the form  is not homogeneous,

dx Ax  By C
but we can formulate it so by a transformation in the origins of x and y.

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