Solve differential equation 6(y-x^2)dx+xdy=0

Solve differential equation 6(y-x^2)dx+xdy=0

Question
Solve differential equation \(6(y-x^2)dx+xdy=0\)

Answers (1)

2020-11-24
\(6(y-x^2)dx+xdy=0\)
\((6(y-x^2)dx)/(xdx)+(xdy)/(xdx)=0\)
\((6(y-x^2)dx)/(xdx)+(xdy)/(xdx)=0\)
\(6((y-x^2)/x)+dy/dx=0\) \(dy/dx+(6/x)y=6x\) (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 \(yx^6=(3x^8)/4+1/4\)
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