Question 10 - CBSE Class 12 Sample Paper for 2018 Boards
Last updated at Sept. 14, 2018 by Teachoo
Verify that ax
2
+ by
2
= 1 is a solution of the differential equation x(yy
2
+ y
1
2
) = yy
1
This is a question of CBSE Sample Paper - Class 12 - 2017/18.
You can download the question paper here
https://www.teachoo.com/cbse/sample-papers/
Subscribe to our Youtube Channel - https://you.tube/teachoo
Transcript
Question 10
Verify that ax2 + by2 = 1 is a solution of the differential equation x(yy2 + y12) = yy1
Given ax2 + by2 = 1
First we find y1, and y2
Now,
ax2 + by2 = 1
Differentiating w.r.t. x
(ax2)โ+ (by2)โ = (1)โ
2ax + 2by ๐๐ฆ/๐๐ฅ = 0
2ax + 2byy1 = 0
2(ax + byy1) = 0
ax + byy1 = 0
Now, finding y2
From (1)
ax + byy1 = 0
ax + by๐๐ฆ/๐๐ฅ = 0
Differentiating w.r.t. x
(ax)โ + ("by" ๐๐ฆ/๐๐ฅ)^โฒ= 0
a + b("y" ๐๐ฆ/๐๐ฅ)^โฒ= 0
a + b(๐ฆ^โฒ ๐๐ฆ/๐๐ฅ+๐ฆ๐ฆโฒโฒ)= 0
a + b(๐ฆ^โฒ ๐ฆโฒ+๐ฆ๐ฆโฒโฒ)= 0
a + b(๐ฆ1 ๐ฆ1+๐ฆ๐ฆ2)= 0
a + b(๐ฆ1 ๐ฆ1+๐ฆ๐ฆ2)= 0
a + b(ใ๐ฆ_1ใ^2+๐ฆ๐ฆ2)= 0
a = โ b(ใ๐ฆ_1ใ^2+๐ฆ๐ฆ2)
Now, from (1)
ax + byy1 = 0
Putting a = โ b(ใ๐ฆ_1ใ^2+๐ฆ๐ฆ2) from (2)
โ b(ใ๐ฆ_1ใ^2+๐ฆ๐ฆ2)x + byy1 = 0
byy1 = b(ใ๐ฆ_1ใ^2+๐ฆ๐ฆ2)x
Cancelling b both sides
yy1 = (ใ๐ฆ_1ใ^2+๐ฆ๐ฆ2)x
x(ใ๐ฆ_1ใ^2+๐ฆ๐ฆ2) = yy1
Hence proved
Show More