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Forming Differential equations
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Forming Differential equations
Last updated at May 29, 2023 by Teachoo
Question 9 Form the differential equation of the family of hyperbolas having foci on π₯βππ₯ππ and center at origin. Equation of hyperbola having foci on x-axis & center at origin (0, 0) is π₯^2/π^2 βπ¦^2/π^2 =1 β΄ Differentiating Both Sides w.r.t. π₯ π/ππ₯ [π₯^2/π^2 βπ¦^2/π^2 ]=π(1)/ππ₯ 1/π^2 [2π₯]β1/π^2 [2π¦ . ππ¦/ππ₯]=0 2π¦/π^2 . π¦β²=2π₯/π^2 Since it has two variables, we will differentiate twice π¦/π^2 π¦β²=π₯/π^2 (π¦/π₯)π¦β²=π^2/π^2 (π¦π¦^β²)/π₯ = π^2/π^2 Again differentiating both sides w.r.t. x ((π¦π¦^β² )^β² π₯ β (ππ₯/ππ₯)(π¦π¦^β² ))/π₯^2 =0 (π¦π¦^β² )^β² π₯ β (1)(π¦π¦^β² )=πΓπ^π (π¦π¦^β² )^β² π₯ βπ¦π¦^β²=π (ππ^β² )^β² π₯ βπ¦π¦^β²=0 (Using Quotient rule and Diff. of constant is 0) (π^β² π^β²+ππβ²β²)π₯ βπ¦π¦^β²=0 (γπ¦^β²γ^2+π¦π¦β²β²)π₯ βπ¦π¦^β²=0 π₯γπ¦^β²γ^2+π₯π¦π¦^β²β²βπ¦π¦^β²=0 πππ^β²β²+πγπ^β²γ^πβππ^β²=π (Using Product rule)