Solve the homogeneous problems
(x²xy+y²) dx-x²dy=0
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Answer:
(x
2
+3xy+y
2
)dx−x
2
dy=0
x
2
(x
2
+3xy+y
2
)
=
dx
dy
1+3(
x
y
)+(
x
y
)
2
=
dx
dy
let
x
y
=t−−−−(1)
y = xt
diff. w.r.t x
dx
dy
=t+x
dx
dt
(i)⇒1+3(t)+(t)
2
=t+x
dx
dt
t
2
+2t+1=x
dx
dt
x
dx
=
(t+1)
2
dt
integrating both sides
∫
x
dx
=∫
(t+1)
2
dt
ln∣x∣=
(t+1)
−1
+c
ln∣x∣=
(
x
y
+1)+c
−1
ln∣x∣=
x+y
−x
+c
at x = 1, y = 0
lnl=
1+0
−1
+c
0=−1+c⇒c=1
ln∣x∣=
x+y
−x
+1
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