Advanced High-School Mathematics

SECTION 5.5 Differential Equations 311

whereM(x,y) andN(x,y) are both homogeneous of thesame degree.
These are important since they can always be reduced to the form (5.2).
Indeed, suppose that M and N are both homogeneous of degree k.
Then we work as follows:

dy

dx

= −

N(x,y)

M(x,y)

= −

xkN(1,y/x)

xkM(1,y/x)

= −

N(1,y/x)

M(1,y/x)

=F

Çy

x

å

which is of the form (5.2), as claimed. Note that Example 2 above is
an example of a homogeneous first-order ODE.

Exercises
In the following problems, find both the general solution as well as
the particular solution satisfying the initial condition.

1. y′= 2xy^2 , y(0) =− 1


2. yy′= 2x, y(0) = 1


3. 3y^2 y′= (1 +y^2 ) cosx, y(0) = 1


4. 2y′=y(y−2), y(0) = 1


5. xyy′= 2y^2 −x^2 , y(1) = 1


6. y′=

y

x

− 3

Çy

x

å 4 / 3

, y(2) = 1

7. 3xy^2 y′= 4y^3 −x^3 , y(2) = 0