Question

An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation. A differential equation of the form \(\frac{dy}{dx} = \text{F(x,y)}\) is said to be homogeneous if F(x, y) is a homogeneous function of degree zero, whereas a function F(x, y) is a homogeneous function of degree n if \(F(\lambda x,\lambda y) = \lambda ^n F(x,y).\) To solve a homogeneous differential equation of the type \(\frac{dy}{dx} = F(x,y) =g\left(\frac{y}{x}\right)\), we make the substitution y = vx and then separate the variables.

Based on the above, answer the following questions :

(I) Show that (x² – y² ) dx + 2xy dy = 0 is a differential equation of the type \(\frac{dy}{dx} = g\left(\frac{y}{x}\right)\).

(II) Solve the above equation to find its general solution.

Answer 1

(I) Based on the information given, we substitute y = vx in the equation (x² – y² ) dx + 2xy dy = 0

i.e., \(\frac{dy}{dx} = \frac{y^2-x^2}{2xy}\left(\text{as y= vx} \Rightarrow \frac{dy}{dx} = \text{v+x}\frac{dv}{dx} \right)\)

\(\Rightarrow \text {v+x}\frac{dv}{dx} = \frac{(V^2-1)x^2}{2\text{v}\text{x}^2}\)

\(\Rightarrow x \frac{dv}{dx} = \frac{(\text{v}^2-1- 2\text{v}^2)}{2\text{v}}\)

\(\Rightarrow \frac{2\text{v}\ dv}{1+\text{v}^2}+ \frac{dx}{x} = 0 \ \ \text{...(i)}\)

\(\therefore\) The above equation, on substituting y = vx is reduced to a variable separable hence given equation is a differential equation of type 

 \(\frac{dy}{dx} = \text{g}\left(\frac{y}{x}\right)\)

(II) Solving (i) further, we ge

\(\int \frac{\text{2v}\ dv}{1+\text{v}^2}+\int \frac{dx}{x} = 0\)

\(\Rightarrow \text{In} (1+\text{v}^2)+ \text{In x = Inc} \)

\(\Rightarrow \text{x} \left(1+\left(\frac{y}{x}\right)^2\right) = \text{c}\)

\(\Rightarrow \text {x}^2 + \text{y}^2 - \text {cx} = 0\)