Hey there! We receieved your request
Stay Tuned as we are going to contact you within 1 Hour
One of our academic counsellors will contact you within 1 working day.
Click to Chat
1800-5470-145
+91 7353221155
Use Coupon: CART20 and get 20% off on all online Study Material
Complete Your Registration (Step 2 of 2 )
Sit and relax as our customer representative will contact you within 1 business day
OTP to be sent to Change
Illustration 1: The differential equation dy/dx = √(1-y2)/y determines a family of circle with:
1. variable radii and affixed centre at (0, 1)
2. variable radii and affixed centre at (0, -1)
3. fixed radius 1 and variable centres along the x-axis
4. fixed radius 1 and variable centres along the y-axis
Solution: The given equation is
dy/dx = √(1-y2)/y
Taking the terms of y and x on separate sides
y/√(1-y2)dy = dx
Integrating both sides, we get
∫y/√(1-y2) dy = ∫dx
-√(1-y2) = x + c
Hence, we get x2 +y2 + 2cx + c2 - 1 =0
or 1- y2 = (x + c)2
This clearly shows that this differential equation represents a circle of fixed radius 1 and variable centres along x –axis.
Illustration 2:Solve (D2 + 4) y = x sin2x.
Solution:The C.F can be easily obtained as C.F. = c1cos 2x + c2 sin2x
P.I. = 1/ (D2+4). x sin 2x
= {x- 1/(D2+4) .2D}. 1/(D2+4). sin 2x
= {x- 1/ (D2+4). 2D}{-x/4 cos 2x}
= -x2/4 cos 2x + ½.1/(D2+4) (cos 2x- 2x sin 2x)
= -x2/4 .cos 2x + ½ 1/(D2+4) cos 2x – 1/(D2+4) x sin 2x
= -x2/8 cos 2x +1/16 x sin 2x
So, y= c1cos 2x + c2 sin 2x – x2/ 8 cos 2x + 1/16 x sin 2x.
Illustration 3:Solvecos (x+y+1) dx- dy = 0
Solution: Rearranging the terms it can easily be reduced to variable separable.
So, dy / dx= cos (x+y+1)
If we substitute t = x+y+1, we get
dt/ dx = 1+ dy/dx
So, dy/dx = (dt/dx)-1.
Hence the equation becomes, (dt / dx) -1 = cos t
So, dt / dx = 1+ cos t
Now taking the terms of t on one side and of x on the other
dt / (1+cos t) = dx ……(1)
For integrating (1+cos t), first we write it as 2 cos2t/2 using the formula cos 2x = 2 cos2x-1.
Now integrating dt/ 2cos2 (t/2) i.e. ½ sec 2 (t/2) gives tan t/2. So, using this in equation (1) we get
tan t/2 – x = c
Hence, tan [(x+y+1)/2] –x = c.
Get your questions answered by the expert for free
You will get reply from our expert in sometime.
We will notify you when Our expert answers your question. To View your Question