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Home Q & A Board Science-Related Homework Help Mathematics
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rjavier1 rjavier1
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11 years ago
d^2*y/dx^2+y*dy/dx = 0
When I try to solve this I get dy/dx+1/2*y^2=c, c being any constant. Can I go further or is this the answer?
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#1
11 years ago
Substitution problem: let u = dy/dx = y ' , then y " = u(y) u ' (y) and your  DE is now  u u ' + u y = 0. This means  u = o, hence y = constant or u '(y) = -y Integrate both side (with respect to y), set  the answer for u equal to y '(x) = and find y(x). You will have two constants of integration as well having to use partial fractions ( or a table) and there are two answers depending upon the sign of the constant from your first integration.
wrote...
#2
11 years ago
Now take the (1/2)y^2 to the other side and you get a separable equation.
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#3
11 years ago
y'' + y y' = 0 is a nonlinear 2nd order DE

first reduce the order of the equation

let p = dy/dx, then d2y/dx2 = p dp/dy

then the differential equation becomes

p dp/dy + yp = 0

dp/dy + y = 0

dp = -y dy

integrate once with respect to y

p = -(y^2)/2 + (C1^2)/2 = (1/2)(C1^2 - y^2) = dy/dx

the constant (C1^2)/2 was chosen to simply the solution.....it could have been just C1 instead

integrate one more time
y = (-1/6)(x)(x^2 - 3 C1^2)+ C2

where C1 and C2 are constants
wrote...
#4
Staff Member
11 years ago
Please mark as solved.
- Master of Science in Biology
- Bachelor of Science
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