\(6(y-x^2)dx+xdy=0\)

\((6(y-x^2)dx)/(xdx)+(xdy)/(xdx)=0\)

\((6(y-x^2)dx)/(xdx)+(xdy)/(xdx)=0\)

\(6((y-x^2)/x)+dy/dx=0\) \(dy/dx+(6/x)y=6x\) (1)

Now, equation (1) is first order linear differential equation

We know that solution of first order linear differential equation is given by

\(y.(I.F.)= int (I.F.)6xdx\) (2)x

where, \(I.F.= e^(int 6/x dx)\)

\(I.F. = e^(6 int dx/x)\) \(( :' int dx/x= ln x+c )\)

\(= e^(6(ln x))\)

\(= e^(ln x^6)\)

\(= x^6\)

Now, solving equation(2) by using I.F.

\(y(x^6)= int x^6(6x)dx\)

\(= 6 int x^7dx\) \(( :' int x^n dx= (x^(n+1))/(n+1)+c\)

\(= 6(x^8/8)+c\)

\(= (3x^8)/4+c\) (3)

Now, using y(1)= 1 in equation (3)

\(1(1)^6= (3(1)^8)/4+c\)

\(1= 3/4+c\)

\(c= 1-3/4\)

\(c= (4-3)/4\)

\(c= 1/4\)

Now, putting value of c in equation (3) \(yx^6= (3x^8)/4+1/4\)

Hence, solution of given differential equation is \(yx^6=(3x^8)/4+1/4\)

\((6(y-x^2)dx)/(xdx)+(xdy)/(xdx)=0\)

\((6(y-x^2)dx)/(xdx)+(xdy)/(xdx)=0\)

\(6((y-x^2)/x)+dy/dx=0\) \(dy/dx+(6/x)y=6x\) (1)

Now, equation (1) is first order linear differential equation

We know that solution of first order linear differential equation is given by

\(y.(I.F.)= int (I.F.)6xdx\) (2)x

where, \(I.F.= e^(int 6/x dx)\)

\(I.F. = e^(6 int dx/x)\) \(( :' int dx/x= ln x+c )\)

\(= e^(6(ln x))\)

\(= e^(ln x^6)\)

\(= x^6\)

Now, solving equation(2) by using I.F.

\(y(x^6)= int x^6(6x)dx\)

\(= 6 int x^7dx\) \(( :' int x^n dx= (x^(n+1))/(n+1)+c\)

\(= 6(x^8/8)+c\)

\(= (3x^8)/4+c\) (3)

Now, using y(1)= 1 in equation (3)

\(1(1)^6= (3(1)^8)/4+c\)

\(1= 3/4+c\)

\(c= 1-3/4\)

\(c= (4-3)/4\)

\(c= 1/4\)

Now, putting value of c in equation (3) \(yx^6= (3x^8)/4+1/4\)

Hence, solution of given differential equation is \(yx^6=(3x^8)/4+1/4\)