Question

For the differential equation in given question find a particular solution satisfying the given condition.​​

$x(x^2-1)\frac{dy}{dx}=1,wherey=0andx=2$

Answer 1

Given, $x(x^2-1)\frac{dy}{dx}=1$
On separating the variables, we get
$dy=\frac{1}{x(x^2-1)}dx\Rightarrow dy=\frac{1}{x(x-1)(x+1)}dx$
On integrating both sides, we get $\int dy=\int \frac{1}{x(x-1)(x+1)}dx....\left(i\right)Let\frac{1}{x(x-1)(x+1)}=\frac{A}{x}+\frac{B}{x-1}+\frac{C}{x+1}....\left(ii\right)$ (by partial fraction)
$\Rightarrow \frac{1}{x(x-1)(x+1)}=\frac{A(x-1)(x+1)+Bx(x+1)+Cx(x-1)}{x(x-1)(x+1)}$

$\therefore 1=A(x^2-1)+B\left(x\right)(x+1)+C\left(x\right)(x-1)\Rightarrow 1=A{x}^2-A+B{x}^2+Bx+C{x}^2-Cx\Rightarrow 1=x^2(A+B+C)+x(B-C)+(-A)$
On comparing the coefficients of $x^2,x$ and constant terms on both sides, we get
A + B + C = 0 ...... (iii)
B - C = 0 ...... (iv)
and -A = 1 ...... (v)
On putting the value of A in Eq. (iii), we get
B + C = 1 ..... (vi)
On adding Eqs. (iv) and (vi), we get $2B=1\Rightarrow B=\frac{1}{2}$
Then, from Eq. (iv), $C=\frac{1}{2}$ and $B=\frac{1}{2}$
On putting the values of A, B and C in Eq. (ii), we get
$\frac{1}{x(x-1)(x+1)}=\frac{-1}{x}+\frac{1}{2(x-1)}+\frac{1}{2(x+1)}\Rightarrow \int dy=-\int \frac{1}{x}dx+\frac{1}{2}\int \frac{1}{x-1}dx+\frac{1}{2}\int \frac{1}{x+1}dx$
Now from eq. (i) we get
$y=-log|x|+\frac{1}{2}log|(x-1)|+\frac{1}{2}log|(x-1)|+C\Rightarrow y=-\frac{1}{2}log\left(x^2\right)+\frac{1}{2}log|(x-1)|+\frac{1}{2}log|(x+1)|+C\Rightarrow y=\frac{1}{2}\left[\left(\frac{(x-1)(x+1)}{x^2}\right)\right]+C(\therefore logl+logm-logn=log\frac{lm}{n})$
$\Rightarrow y=\frac{1}{2}\left[log\left|\left(\frac{x^2-1}{x^2}\right)\right|\right]+C$ ...... (vii)
On putting y = 0 and x = 2 in Eq. (vii), we get
$\therefore 0=\frac{1}{2}\left[log\left|\left(\frac{2^2-1}{2^2}\right)\right|\right]+C\Rightarrow C=-\frac{1}{2}log\left(\frac{3}{4}\right)$
On substituting the value of C in Eq. (vii), we get
$y=\frac{1}{2}log\left|\left(\frac{x^2-1}{x^2}\right)\right|-\frac{1}{2}log\left(\frac{3}{4}\right)$
which is the required particular solution.