Solve the following differential equation, hence find the particular solution when \(x=0,\ y=1\)\(y^3-\frac{dy}{dx}=x\frac{dy}{dx}\)Solution:\(y^3=x\frac{dy}{dx}+\frac{dy}{dx}\)\(\therefore y^3=(x+1)\)[ ]\(\therefore (x+1)\,dy=y^3dx\)Separating the v…

Q.3 (C) (ii) Solve the following differential equation, hence find the particular solution when \(x=0,\ y=1\)
\(y^3-\frac{dy}{dx}=x\frac{dy}{dx}\)

Solution:
\(y^3=x\frac{dy}{dx}+\frac{dy}{dx}\)
\(\therefore y^3=(x+1)\)[ ]
\(\therefore (x+1)\,dy=y^3dx\)
Separating the variables, we get
\(\frac{1}{y^3}dy=\frac{1}{x+1}dx\)
Now, integrating, we get
\(\therefore\int\frac{1}{y^3}dy=\int\frac{1}{x+1}dx\)
\(\therefore -\frac{1}{2y^2}=\)[ ]\(+c\) ....(I)
which is required general solution
put \(x=0\) and \(y=1\) in (I)
\(-\frac{1}{2(1)^2}=\log|0+1|+c\)
\(\therefore\)[ ]\(=c\)
\(\therefore -\frac{1}{2y^2}=\)[ ]\(-\frac{1}{2}\)
is the particular solution.