Easy separable differential equation with confusing initial condition?
dy/dx = (y-1)^2 + 0.01 y(0) = 1
1. Solving the DE by separation method yields...
1/[(y-1)^2 + 0.01)] * dy = dx
letting u = y-1 gives an integral solution with inverse tangent like this....
1/y-1 *arctan(1/10(y-1)) = x + C
Plugging in the initial condition makes the equation undefined. Does anyone know how to go about this. The answer from the back of the book is...
y = 1 + 1/10*tan(1/10x)
Not sure how you are able to get a value for C here...
Thanks for any help!
- az_lenderLv 78 years agoFavorite Answer
1. If I start from the "y" given at the back of the book, I find
the given answer does satisfy both the differential equation
and the initial condition, so I'm going to check over your work.
dy/dx = (y-1)^2 + 0.01
dy/ [ (y-1)^2 + 0.01 ] = dx
y-1 = 0.1 tan(u)
(y-1)^2 = 0.01 tan^2(u)
dy = 0.1 sec^2(u) du
The equation now becomes
[0.1 sec^2(u)] du / [0.01 sec^2(u)] = dx
du = (1/10) dx
u = (1/10)x + C
arctan[10(y-1)] = (1/10)x + C
10(y-1) = tan [ (1/10)x + C ]
Since y(0) = 1, you have
10(0) = tan(0+C), which implies that C=0.
Nothing here is "undefined."
10(y-1) = tan[(1/10)x]
y = 1 + (1/10)tan[(1/10)x]
- zulaufLv 44 years ago
For the fundamental i'm getting fundamental((one hundred/((10*(y - a million))^2 + a million)) dy). Then I permit u = 10*(y - a million); so du = 10 dy and the fundamental turns into 10*fundamental((a million/(u^2 + a million)) du) = 10*arctan(u) + C = 10*arctan(10*(y - a million)) + C. So we've 10*arctan(10*(y - a million)) = x + C ----> arctan(10*(y - a million)) = (a million/10)*x + C -----> 10*(y - a million) = tan((a million/10)*x + C) ----> y = a million + (a million/10)*tan((a million/10)*x + C). Now prepare the preliminary subject to discover that y(0) = a million + (a million/10)*tan((a million/10)*0 + C) = a million, and to that end we require that tan(C) = 0. it incredibly is the case for C = n*pi for any integer n. So y = a million + (a million/10)*tan((a million/10)*x + n*pi). Edit: comments: (i) i exploit the consistent C in a usual experience; it incredibly is authorized to "morph" from one step to the subsequent until the preliminary subject is utilized; (ii) shall we merely permit n = 0 and simplify the answer to y = a million +(a million/10)*tan((a million/10)*x) for sake of readability. :)