Find the general solution of the differential equation: $\left(xy-x^{2}\right)dy=y^{2}\,dx$

$Cy=e^{y/x}$ (equivalently $\dfrac{y}{x}=\log|Cy|$)

CBSE Class 12 · 2023 · Differential EquationsLong Answer

Find the general solution of the differential equation:

(x y - x^{2}) d y = y^{2} d x

Answer: $C y = e^{y / x}$ (equivalently $\frac{y}{x} = lo g ∣ C y ∣$ )

Write

\frac{d y}{d x} = \frac{y ^{2}}{x y - x ^{2}}

(homogeneous). Put

y = v x

v + x \frac{d v}{d x} = \frac{v ^{2}}{v - 1} \Rightarrow x \frac{d v}{d x} = \frac{v}{v - 1}

Separate:

\frac{v - 1}{v} d v = \frac{d x}{x} \Rightarrow \int (1 - \frac{1}{v}) d v = \int \frac{d x}{x}

v - lo g ∣ v ∣ = lo g ∣ x ∣ + lo g C \Rightarrow v = lo g ∣ C xv ∣

With

v = \frac{y}{x}

(so

xv = y

\frac{y}{x} = lo g ∣ C y ∣ \Rightarrow C y = e^{y / x}

CBSE Class 12 Board · 2023 · Subjective