1. Chapter 9 Class 12 Differential Equations
2. Serial order wise

Transcript

Example 22 Find the particular solution of the differential equation ππ¦/ππ₯+π¦ cotβ‘γπ₯=2π₯+π₯^2 cotβ‘π₯(π₯β 0) γ given that π¦=0 π€βππ π₯=π/2 ππ¦/ππ₯+π¦ cotβ‘γπ₯=2π₯+π₯^2 cotβ‘π₯ γ Differential equation is of the form ππ¦/ππ₯+ππ¦=π where P = cot x & Q = 2x + x2 cot x IF = π^β«1βγπ ππ₯γ IF = π^β«1βγcotβ‘π₯ ππ₯γ IF = π^"log sin x " IF = sin x Solution is y (IF) =β«1βγ(πΓπΌπΉ) ππ₯+πγ y sin x β«1βγsinβ‘π₯=(2π₯+π₯^(2 ) cotβ‘π₯ ) ππ₯+πγ y sin x = β«1βγ(2π₯ sinβ‘π₯+π₯^(2 ) sinβ‘γπ₯ cotβ‘π₯ γ ) ππ₯+πγ y sinβ‘π₯ = β«1βγ2π₯ sinβ‘π₯ ππ₯+γ β«1βγπ₯^2 sinβ‘π₯ cotβ‘π₯ ππ₯+γ πΆ y sinβ‘π₯ = 2β«1βγsinβ‘π₯ (π₯)ππ₯+ππ₯^2 sinβ‘π₯ cotβ‘γπ₯ ππ₯+πγ γ y sinβ‘π₯ = 2 [sinβ‘γπ₯ β«1βγπ₯ ππ₯βγ β«1βγ[cosβ‘γπ₯ β«1βγπ₯ ππ₯γ γ ] ππ₯γ γ ] + β«1βγπ₯^2 sinβ‘π₯ γ cosβ‘π₯/sinβ‘π₯ dx + C y sin x = 2 sin x ["sin x " [π₯^2/2]" β2" β«1βγγcos xγβ‘[π₯^2/2] ππ₯+β«1βπ₯^2 γ cosβ‘π₯ " dx + c" ] y sin x = x2sin x β β«1βπ₯^2 cos x dx + β«1βπ₯^2 cos x dx + c y sin x = x2 sin x + c Given that y = 0 when x = π/2 Putting π₯=π/2 and y = 0 in (1) (0) sin π/2=(π/2)^2 sinβ‘γ(π/2)+Cγ 0 =π^2/4 (1)+C γβπγ^2/4=C Putting value in C in (1) y sin x = π₯^2 sinβ‘γπ₯ βγ π^2/4 Dividing whole by sin x (π¦ sinβ‘π₯)/sinβ‘π₯ =(π₯^2 sinβ‘π₯)/sinβ‘π₯ βπ^2/(4 sinβ‘π₯ ) π=π^πβπ^π/γπ π¬π’π§γβ‘π