Question

Form the differential equation by eliminating A and B in Ax² + By² = 1

Answer 1

Given that Ax² + By² = 1

Differentiating w.r.t. x, we get

`2"A"  . x + 2"B"y "dy"/"dx"` = 0

⇒ `"A"x + "B"y  . "dy"/"dx"` = 0

⇒ `"B"y . "dy"/"dx"` = –Ax

∴ `y/x * "dy"/"dx" = - "A"/"B"`

Differentiating both sides again w.r.t. x, we have

`y/x * ("d"^2y)/("d"x^2) + "dy"/"dx"((x * "dy"/"dx" - y.1)/x^2)` = 0

⇒ `(yx^2)/x * ("d"^2y)/("d"x^2) + x * ("dy"/"dx")^2 - y * "dy"/"dx"` = 0

⇒ `xy * ("d"^2y)/("d"x^2) + x * ("dy"/"dx")^2 - y * "dy"/"dx"` = 0

⇒ `xy * y"''" + x*(y"'")^2 - y*y"'"` = 0

Hence, the required equation is `xy * y"''" + x*(y"'")^2 - y*y"'"` = 0