Differential Equations — Class 12 Maths Solution

: $y' = \cfrac{{x + y}}{x}\,\,{\rm{or}}\,\,\cfrac{{dy}}{{dx}} = 1 + \cfrac{y}{x}$

…(1)
Since R.H.S. is of the form $g(y/x)$ ,

so it is a homogeneous function of degree zero.

Therefore, equation (1) is a homogeneous differential equation.

To solve this, we put $y = vx$

…(2)
Differentiating (2) w.r.t. $x$,

we get $\cfrac{{dy}}{{dx}} = v + x\cfrac{{dv}}{{dx}}$

Substituting the value of $y$ and $\cfrac{{dy}}{{dx}}$ in equation (1),

we get
$v + x\cfrac{{dv}}{{dx}} = 1 + v \Rightarrow \cfrac{{dv}}{{dx}} = \cfrac{1}{x} \Rightarrow dv = \cfrac{{dx}}{x}$

Integrating,

we get $v = \log |x| + C$
$\Rightarrow \cfrac{y}{x} = \log \left| x \right| + C \Rightarrow y = x\log \left| x \right| + Cx$

which is the required general solution.