# Calculus

How do you calculate the integral of xy' = y + √(x^2+y^2) as a homogeneous differential equation?

1. 👍 0
2. 👎 0
3. 👁 100
1. First, let v = y/x
Now the equation is
y' = v + √(1+v^2)
Now, what about y'? That is dy/dx, but now we have a bunch of v stuff. Luckily,
y = vx
y' = xv' + v
Now we have
xv'+v = v + √(1+v^2)
xv' = √(1+v^2)
dv/√(1+v^2) = dx/x
arcsinh(v) = lnx+c
v = sinh(lnx+c)
Now, v=y/x, so
y = x sinh(lnx+c)
That's pretty simple, but we can get rid of the sinh at the cost of a messier answer.
Now, recall that sinh(z) = (e^z - e^-z)/2 so
y = x (e^(lnx+c) - e^-(lnx+c))/2
Now, since e^lnx = x, e^(lnx+c) = xe^c
y = x(xe^c - 1/(xe^c))/2
= x(x^2e^2c-1)/(2xe^c)
= ((xe^c)^2-1)/(2e^c)
I think I prefer the sinh form.

1. 👍 0
2. 👎 0
posted by oobleck

## Similar Questions

1. ### science

Will you please check my answers? Substances or mixtures? 1. Carbon Dioxide = M 2.salt water =M 3. table salt = M Homogeneous or Heterogeneous Matter- 1. flat soda pop = Heterogeneous 2. aluminum foil = Homogeneous 3. black coffee

asked by Reed on September 12, 2012
2. ### Calculus

Hello, I have some calculus homework that I can't seem to get started..at least not on the right track? I have 3 questions 1. integral of [(p^5)*(lnp)dp] I'm using the uv-integral v du formula So first, I'm finding u and I think

asked by Shelley on January 23, 2007
3. ### Math

I need to solve this set of differential equations {y'+(t/2)1, y(0)=1} and then find the limiting value as t approaches infinity. I'm not sure where to start. I found the answer to the differential equation to be:

asked by Emily on September 6, 2012

Choose either homogeneous or heterogeneous. Tea: homogenous Concrete: heterogeneous Soda: homogeneous Soapy Water: homogeneous Salt Water: homogeneous Granite: heterogeneous Please help, and thanks lots :D

asked by Anonymous on March 3, 2015
5. ### mathematics

what does 'dx' signify in the integral and differentiation notation? I was reading integration by substitution method where it is required to replace 'dx' and change the independent variable such that the integral could be

asked by Arvind on December 26, 2013
6. ### Calculus URGENT test tonight

Integral of: __1__ (sqrt(x)+1)^2 dx The answer is: 2ln abs(1+sqrt(x)) + 2(1+sqrt(X))^-1 +c I have no clue why that is! Please help. I used substitution and made u= sqrt(x)+1 but i don't know what happened along the way! Your first

asked by Gabriela on April 20, 2007
7. ### calc

how do you start this problem: integral of xe^(-2x) There are two ways: 1) Integration by parts. 2) Differentiation w.r.t. a suitably chosen parameter. Lets do 1) first. This is the "standard method", but it is often more tedious

asked by walex on May 15, 2007
8. ### integration by parts

s- integral s ln (2x+1)dx ? = ln(2x+1)x - s x d( ln (2x+1)) = ln(2x+1)x- s x [(2x+1)'/ (2x+1)] dx = ln(2x+1)x- s x [(2)/ (2x+1)] ?... then i'm confused... "ln(2x+1)x- s x [(2)/ (2x+1)] ?... then i'm confused..." x [(2)/ (2x+1)] =

asked by nicholas on February 17, 2007
9. ### calc asap!

can you help me get started on this integral by parts? 4 S sqrt(t) ln(t) dt 1 please help! thanks! Integral t^(1/2)Ln(t)dt = 2/3 t^(3/2)Ln(t)- 2/3 Integral t^(1/2) dt = 2/3 t^(3/2)Ln(t) - 4/9 t^(3/2) Simpler method: Integral

asked by Woof on June 7, 2007
10. ### physics

State only Gauss law in integral form.Use it to find out electric field due to an isolated homogeneous sphere of charges at an outside point.

asked by mania on February 10, 2013

More Similar Questions