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Byju's Answer
Standard XII
Mathematics
Formation of a Differential Equation from a General Solution
Solve : x 2...
Question
Solve :
x
2
y
d
x
=
(
x
3
+
y
3
)
d
y
=
0
.
Open in App
Solution
The given homogeneous differential equation is ....
if we assume y/x = t
y = xt
dy/dx = t + xdt/dx
then, t/(1 + t³) = t + x.dt/dx
t/(1 + t³) - t = xdt/dx
(t - t - t⁴)/(1 + t³) = x dt/dx
-t⁴/(1 + t³) = x dt/dx
dx/x = -(1 + t³)/t⁴ dt
lnx = -t^-4 dt - dt/t
lnx = -t^-3/-3 - lnt + C
lnx = 1/3t³ - lnt + C
now, put t = y/x
so, lnx = 1/3(y/x)³ - lny/x + C
or, lnx = x³/3y³ - lny + lnx + C
hence, -x³/3y³ + lny + C = 0
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