Question

The curves satisfying the differential equation (1 – x²) y¢ + xy = ax are –

• A. Ellipses and hyperbolas
• B. Ellipses and parabola
• C. Ellipses and straight lines
• D. Circles and ellipses

Answer 1

Answer: (A)

Ellipses and hyperbolas

Answer 2

The given equation is linear in y and can be written as

$\frac{dy}{dx}$ + $\frac{x}{1 - x^{2}}$y = $\frac{ax}{1 - x^{2}}$

Its integrating factor is

$e^{\int_{}^{}{\frac{x}{1 - x^{2}}dx}}$= $e^{–(1/2)\log(1–x^{2})}$= $\frac{1}{\sqrt{1 - x^{2}}}$if –1 < x < 1

and if x² > 1 then I.F. = $\frac{1}{\sqrt{x^{2} - 1}}$

$\frac { \mathrm { d } } { \mathrm { dx } }$ $\left( y\frac{1}{\sqrt{1 - x^{2}}} \right)$ = $\frac{ax}{(1 - x^{2})^{3/2}}$ = –$\frac{1}{2}$a $\frac{- 2x}{(1 - x^{2})^{3/2}}$

Žy$\frac{1}{\sqrt{1 - x^{2}}}$ =$\frac{a}{\sqrt{1 - x^{2}}}$ + C

Ž y = a + C $\sqrt{1 - x^{2}}$

Ž (y – a)² = C²(1 – x²) Ž (y – a)² + C² x² = C²

Thus if – 1 < x < 1 the given equation represents an ellipse. If x² > 1 then the solution is of the form – (y – a)² – C² x² = C² which represents a hyperbola