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 March 22, 2023 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

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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