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