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Q.
The general solution of the homogeneous differential equation of the type
$\frac{d y}{d x}=F(x, y)=g\left(\frac{y}{x}\right)$, when $y=v \cdot x$ is
Differential Equations
Solution:
To solve a homogeneous differential equation of the type
$\frac{d y}{d x}=F(x, y)=g\left(\frac{y}{x}\right) ....$(i)
We make the substitution
$y=v \cdot x ....$(ii)
On differentiating Eq. (ii) w.r.t. $x$, we get
$\frac{d y}{d x}=v+x \frac{d v}{d x} ....$(iii)
On substituting the value of $\frac{d y}{d x}$ from Eq. (iii) in Eq. (i), we get
$v+x \frac{d v}{d x}=g(v)$
or $ x \frac{d v}{d x}=g(v)-v ....$(iv)
On separating the variables in Eq. (iv), we get
$\frac{d v}{g(v)-v}=\frac{d x}{x} ....$(v)
On integrating both sides of Eq. (v), we get
$\int \frac{d v}{g(v)-v}=\int \frac{1}{x} d x+C .....$(vi)
Eq. (vi) gives general solution (primitive) of the differential Eq. (i) when we replace $v$ by $\frac{v}{x}$.
Note If the homogeneous differential equation is in the form $\frac{d x}{d y}=F(x, y)$ where, $F(x, y)$ is homogeneous function of degree zero, then we make substitution $\frac{x}{y}=v$, i.e., $x=v y$ and we proceed further to find the general solution as discussed above by writing $\frac{d x}{d y}=F(x, y)=h\left(\frac{x}{y}\right)$