Forming Differential equations

Chapter 9 Class 12 Differential Equations
Serial order wise

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Question 2 Form a differential equation representing the given family of curves by eliminating arbitrary constants π and π. π¦^2=π(π^2βπ₯^2 ) π¦^2=π(π^2βπ₯^2 ) π¦^2=ππ^2βππ₯^2 Since it has two variables, we will differentiate twice β΄ Diff. Both Sides w.r.t. π₯ 2π¦.ππ¦/ππ₯=0β2ππ₯ 2π¦π¦^β²=β2ππ₯ π¦π¦β²=βππ₯ (π¦π¦^β²)/(βπ₯) = π π = (βπ¦)/π₯ π¦β² Now, π¦π¦β²=βππ₯ "Again Differentiating w.r.t. " π₯ ππ¦/ππ₯.π¦^β²+π¦.(π(π¦^β²))/ππ₯=βπ ππ₯/ππ₯ π¦^β²Γπ¦^β²+π¦Γπ¦^β²β²=βπ γπ¦^β²γ^2+π¦π¦^β²β²=β((βπ¦)/π₯ π¦β²) γπ¦^β²γ^2+π¦π¦^β²β²=(π¦π¦^β²)/π₯ π₯γπ¦^β²γ^2+π₯π¦π¦^β²β²=π¦π¦^β² β¦(1) ("Using Product Rule ") (From (1) π= (βπ¦)/π₯ ππ¦/ππ₯ ) πππ^β²β²+πγπ^β²γ^πβππ^β²=π which is the required differential equation