Forming Differential equations

Chapter 9 Class 12 Differential Equations
Serial order wise

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Question 3 Form a differential equation representing the given family of curves by eliminating arbitrary constants π and π. π¦=π π^3π₯+π π^(β2π₯) Since it has two variables, we will differentiate twice π¦=π π^3π₯+π π^(β2π₯) β΄ Differentiating Both Sides w.r.t. π₯ ππ¦/ππ₯=π/ππ₯ [ππ^3π₯+π π^(β2π₯) ] =ππ^3π₯Γ3+π π^(β2π₯)Γ(β2) =3ππ^3π₯β2π π^(β2π₯) β΄ π¦^β²=3ππ^3π₯β2π π^(β2π₯) ...(1) π¦^β²=3ππ^3π₯β2π π^(β2π₯) Again differentiating w.r.t. π₯ π¦^β²β²=π/ππ₯ [3ππ^3π₯β2π π^(β2π₯) ] π¦^β²β²=3ππ^3π₯ (3)β2π π^(β2π₯) (β2) β΄ π¦^β²β²=9ππ^3π₯+4π π^(β2π₯) Subtracting (2) From (1) π¦^β²β²βπ¦^β²=9ππ^3π₯+4π π^(β2π₯)β3ππ^3π₯+2π π^(β2π₯) π¦^β²β²βπ¦^β²=6ππ^3π₯+6π π^(β2π₯) π¦^β²β²βπ¦^β²=6(ππ^3π₯+ππ^(β2π₯)) π¦^β²β²βπ¦^β²=6y π^β²β²βπ^β²βππ=π is the required differential equation. (As y = π^3π₯ + bπ^3π₯)