(${\mathit{x}}^2\mathit{–}{\mathit{y}}^2$)dx + 2xydy = 0
⇒ $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = $\frac{{\mathit{y}}^2-{\mathit{x}}^2}{2\mathit{x}\mathit{y}}$ ... (1)
It is a homogeneous differential equation.
Let y = vx ...(2)
∴ $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = v + x $\frac{\mathit{d}\mathit{v}}{\mathit{d}\mathit{x}}$ ... (3)
Substituting (2) and (3) in (1), we get:
v + x $\frac{\mathit{d}\mathit{v}}{\mathit{d}\mathit{x}}$ = $\frac{{\mathit{v}}^2{\mathit{x}}^2-{\mathit{x}}^2}{2\mathit{x}\mathit{v}\mathit{x}}$
v + x $\frac{\mathit{d}\mathit{v}}{\mathit{d}\mathit{x}}$ = $\frac{{\mathit{x}}^2{\mathit{v}}^2-1}{2\mathit{v}{\mathit{x}}^2}$ - $\frac{{\mathit{v}}^2-1}{2\mathit{v}}$
$2{\mathit{v}}^2+2\mathit{v}\mathit{x}\frac{\mathit{d}\mathit{v}}{\mathit{d}\mathit{x}}$ = ${\mathit{v}}^2$ - 1
2vx $\frac{\mathit{d}\mathit{v}}{\mathit{d}\mathit{x}}$ = - ${\mathit{v}}^2$ - 1
$\left(\frac{2\mathit{v}}{{\mathit{v}}^2+1}\right)$ dv = - $\frac{\mathit{d}\mathit{x}}{\mathit{x}}$
Integrating both sides, we get:
∫ $\frac{2\mathit{v}}{{\mathit{v}}^2+1}$ dv = - ∫ $\left(\frac{1}{\mathit{x}}\right)$ dx
log $\left|{\mathit{v}}^2+1\right|$ = - log |x| + log C
log $\left|{\mathit{v}}^2+1\right|$ = log $\left|\frac{\mathit{C}}{\mathit{x}}\right|$
${\mathit{v}}^2$ + 1 = $\frac{\mathit{C}}{\mathit{x}}$
$\mathit{x}{\mathit{v}}^2$ + 1 = C
x $\left|{\left(\frac{\mathit{y}}{\mathit{x}}\right)}^2+1\right|$ = C
${\mathit{y}}^2+{\mathit{x}}^2$ = Cx ... 4
It is given that when x = 1, y = 1
${\left(1\right)}^2+{\left(1\right)}^2$ = C(1)
C = 2
Thus, the required solution is ${\mathit{y}}^2+{\mathit{x}}^2$ = 2x.
OR
We need to solve the following differential equation
$\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = $\frac{\mathit{x}\left(2\mathit{y}-\mathit{x}\right)}{\mathit{x}\left(2\mathit{y}+\mathit{x}\right)}$
$\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = $\frac{2\mathit{y}-\mathit{x}}{2\mathit{y}+\mathit{x}}$ ... (1)
It is a homogeneous differential equation.
Let y = vx ...(2)
∴ $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = v + x $\frac{\mathit{d}\mathit{v}}{\mathit{d}\mathit{x}}$ ... 3
Substituting (2) and (3) in (1), we get:
v + x $\frac{\mathit{d}\mathit{v}}{\mathit{d}\mathit{x}}$ = $\frac{\mathit{x}\left(2\mathit{v}-1\right)}{\mathit{x}\left(2\mathit{v}+1\right)}$
x $\frac{\mathit{d}\mathit{v}}{\mathit{d}\mathit{x}}$ = $\frac{\mathit{v}-1}{2\mathit{v}+1}$ - v
x $\frac{\mathit{d}\mathit{v}}{\mathit{d}\mathit{x}}$ = $\frac{-2{\mathit{v}}^2+\mathit{v}-1}{2\mathit{v}+1}$
$\left(\frac{2\mathit{v}+1}{-2{\mathit{v}}^2+\mathit{v}-1}\right)$ dv = $\left(\frac{1}{\mathit{x}}\right)$ dx
$\left(\frac{2\mathit{v}+1}{2{\mathit{v}}^2-\mathit{v}+1}\right)$ dv = $\left(-\frac{1}{\mathit{x}}\right)$ dx
Integrating both sides,
$\int \frac{1}{2}\left(\frac{4\mathit{v}-1+3}{2{\mathit{v}}^2-\mathit{v}+1}\right)$ dv = ∫ $\left(-\frac{1}{\mathit{x}}\right)$ dx
∫ $\frac{1}{2}\left(\frac{4\mathit{v}-1}{2{\mathit{v}}^2-\mathit{v}+1}\right)$ dv + ∫ $\frac{3}{2}\left(\frac{1}{2{\mathit{v}}^2-\mathit{v}+1}\right)$ dv = ∫ $\left(-\frac{1}{\mathit{x}}\right)$ dx
∫ $\frac{1}{2}\left(\frac{4\mathit{v}-1}{2{\mathit{v}}^2-\mathit{v}+1}\right)$ dv + ∫ $\frac{3}{4}|\frac{1}{{\mathit{v}}^2-\frac{\mathit{v}}{2}+\frac{1}{2}}|$ dv = f $\left(-\frac{1}{\mathit{x}}\right)$ dx
1/2 log $\left|2{\mathit{v}}^2-\mathit{v}+1\right|$ + $\frac{3}{4}\int \left(\frac{1}{{\mathit{v}}^2-\frac{\mathit{v}}{2}+\frac{1}{16}+\frac{7}{16}}\right)$ dv = - log |x| + C
1/2 log $\left|2{\mathit{v}}^2-\mathit{v}+1\right|$ + $\frac{3}{4}$ ∫ $\frac{\mathit{d}\mathit{v}}{{\left(\mathit{v}-\frac{1}{4}\right)}^2+{\left(\frac{\sqrt{7}}{4}\right)}^2}$ = - log |x| + C
1/2 log $\left|2{\mathit{v}}^2-\mathit{v}+1\right|$ + $\frac{3}{4}\times \frac{4}{\sqrt{7}}\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{\mathit{v}-\frac{1}{4}}{\frac{\sqrt{7}}{4}}\right)$ = - log |x| + C
1/2 log $\left|2{\mathit{v}}^2-\mathit{v}+1\right|$ + $\frac{3}{\sqrt{7}}\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{4\mathit{v}-1}{\sqrt{7}}\right)$ = C - log |x|
Put v = $\frac{\mathit{y}}{\mathit{x}}$
$\frac{1}{2}$ log $\left|2{\left(\frac{\mathit{y}}{\mathit{x}}\right)}^2-\left(\frac{\mathit{y}}{\mathit{x}}\right)+1\right|$ + $\frac{3}{\sqrt{7}}\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{\frac{4\mathit{y}}{\mathit{x}}-1}{\sqrt{7}}\right)$ = C - log |x|
$\frac{1}{2}$ log $\left|\frac{2{\mathit{y}}^2-\mathit{x}\mathit{y}+{\mathit{x}}^2}{{\mathit{x}}^2}\right|$ + $\frac{3}{\sqrt{7}}\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{4\mathit{y}-\mathit{x}}{\sqrt{7}\mathit{x}}\right)$ = C - log |x| ... 4
Now y = 1 when x = 1
$\frac{1}{2}$ log $\left|\frac{\left(2\right){\left(1\right)}^2-1\left(1\right)+1^2}{1^2}\right|$ + $\frac{3}{\sqrt{7}}\mathit{t}\mathit{a}{\mathit{n}}^{-1}|\frac{\left(4\right)\left(1\right)-1}{\sqrt{7}\left(1\right)}|$ = C - log |1|
$\frac{1}{2}$ log 2 + $\frac{3}{\sqrt{7}}\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{3}{\sqrt{7}}\right)$ = C ... (5)
Therefore, form (4) and (5) we get:
$\frac{1}{2}$ log $\left|\frac{2{\mathit{y}}^2-\mathit{x}\mathit{y}+{\mathit{x}}^2}{{\mathit{x}}^2}\right|$ + $\frac{3}{\sqrt{7}}\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{4\mathit{y}-\mathit{x}}{\sqrt{7}\mathit{x}}\right)$ = $\frac{1}{2}$ log 2 + $\frac{3}{\sqrt{7}}\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{3}{\sqrt{7}}\right)$ - log |x|
$\frac{1}{2}$ log $\left|\frac{2{\mathit{y}}^2-\mathit{x}\mathit{y}+{\mathit{x}}^2}{{\mathit{x}}^2}\right|$ - $\frac{1}{2}$ log 2 + log |x| = $\frac{3}{\sqrt{7}}\left[\mathit{t}\mathit{a}{\mathit{n}}^{-1}\frac{3}{\sqrt{7}}-\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{4\mathit{y}-\mathit{x}}{\sqrt{7}\mathit{x}}\right)\right]$
$\frac{1}{2}$ log $\left|\frac{2{\mathit{y}}^2-\mathit{x}\mathit{y}+{\mathit{x}}^2}{2{\mathit{x}}^2}.{\mathit{x}}^2\right|$ = $\frac{3}{\sqrt{7}}\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{\frac{3\mathit{x}-4\mathit{y}+\mathit{x}}{\sqrt{7}\mathit{x}}}{1+\frac{3\left(4\mathit{y}-\mathit{x}\right)}{7\mathit{x}}}\right)$
$\frac{1}{2}$ log $\left|\frac{2{\mathit{y}}^2-\mathit{x}\mathit{y}+{\mathit{x}}^2}{2}\right|$ = $\frac{3}{\sqrt{7}}\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{\frac{4\left(\mathit{x}-\mathit{y}\right)}{\sqrt{7}\mathit{x}}}{\frac{7\mathit{x}+12\mathit{y}-3\mathit{x}}{7\mathit{x}}}\right)$
$\frac{1}{2}$ log $\left|\frac{2{\mathit{y}}^2-\mathit{x}\mathit{y}+{\mathit{x}}^2}{2}\right|$ = $\frac{3}{\sqrt{7}}\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{\sqrt{7}\left(\mathit{x}-\mathit{y}\right)}{\mathit{x}+3\mathit{y}}\right)$