Question 8 (Choice 2) - CBSE Class 12 Sample Paper for 2022 Boards (For Term 2) - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards
Last updated at Jan. 23, 2022 by Teachoo
Find the particular solution of the following differential equation, given that y = 0 when π₯ = π/4ππ¦/ππ₯+π¦πππ‘π₯ 2/(1 + sinβ‘π₯ )
Question 8 (Choice 2) Find the particular solution of the following differential equation, given that y = 0 when π₯ = π/4 ππ¦/ππ₯+π¦πππ‘π₯= 2/(1 + sinβ‘π₯ ) ππ¦/ππ₯+π¦πππ‘π₯= 2/(1 + sinβ‘π₯ )
Differential equation is of the form
ππ¦/ππ₯+ππ¦=π
where P = cot x & Q = π/(π + πππβ‘π )
Now,
IF = π^β«1βγπ ππ₯γ
IF = π^β«1βγcotβ‘π₯ ππ₯γ
IF = γπ^π₯π¨π β‘π¬π’π§β‘π γ^" "
IF = sin x
Solution is
y (IF) =β«1βγ(πΓπΌπΉ) ππ₯+πγ
y sin x = β«1βγπ/(π + πππβ‘π )Γ πππ π π πγ + C
y sin x = 2β«1βγ(π ππ π₯)/(1 + π ππβ‘π₯ ) ππ₯γ + C
y sin x = 2β«1βγ((1 + π ππ π₯ β 1))/(1 + π ππβ‘π₯ ) ππ₯γ + C
y sin x = 2β«1βγ((1 + π ππ π₯))/(1 + π ππβ‘π₯ ) ππ₯γβ2β«1βγ1/(1 + π ππβ‘π₯ ) ππ₯γ + C
y sin x = πβ«1βπ πβπβ«1βγπ/(π + πππβ‘π ) π πγ + C
y sin x = 2π₯β2β«1βγπ/(π + πππβ‘π ) π πγ + C
y sin x = 2π₯β2β«1βγ1/(1 + π ππβ‘π₯ ) Γ(π β π¬π’π§β‘π)/(π β πππβ‘π ) ππ₯γ + C
y sin x = 2π₯β2β«1βγ(1 β sinβ‘π₯)/(1 β sin^2β‘π₯ ) ππ₯γ + C
y sin x = 2π₯β2β«1βγ(π β πππβ‘π)/γππ¨π¬γ^πβ‘π π πγ + C
y sin x = 2π₯β2[β«1βγ1/γπππ γ^2β‘π₯ ππ₯γββ«1βγsinβ‘π₯/γπππ γ^2β‘π₯ ππ₯γ] + C
y sin x = 2π₯β2[β«1βγsec^2β‘π₯ ππ₯γββ«1βγsinβ‘π₯/(πππ π₯) Γ1/cosβ‘π₯ ππ₯γ] + C
y sin x = 2π₯β2[β«1βγγπππγ^πβ‘π π πγββ«1βγπππ§β‘π π¬ππβ‘π π πγ] + C
y sin x = ππβπ πππ§β‘π+π πππ π + C
We need to find particular solution when y = 0 when π₯ = π/4
Putting y = 0 and π = π /π
0 Γ sin π /π = 2(π /π)β2 tanβ‘γπ /πγ+2 π ππ π /π + C
0 = π/2β2 Γ 1+2β2 + C
2β2β2βπ/2 = C
C = π(πββπ)βπ /π
Thus, our particular solution is
y sin x = ππβπ πππ§β‘π+π πππ π + π(πββπ)βπ /π
Made by
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.