Math 225 Differential Equations Notes Chapter 2

• Is the equation linear
No
• Is it a function of y/x
dy
dx = xy + y2 + x 2
x 2
dy
dx = xy + y2 + x 2
x 2 = y x + (y x )2 + 1
So the equation is homogeneous.
using the substitution we get:
v + x dv
dx = v + v2 + 1
This equation is separable so:
• ∫ ∫ 1 1
v1
2 dv =
x dx
•
•
•
•
arctan(v) = ln|x| + C
v = tan(ln|x| + C)
y
x
= tan(ln|x| + C)
y = xtan(ln|x| + C)
• Also note x ≡ 0 is also a solution.
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