Question

xsqrt(1+y^2)dx-ysqrt(1+x^2)dy=0
solve using variable separable.

Accepted Answer

You need to start separate the variables, hence, you need to move the negative member to the right side, such that:
xsqrt(1 + y^2)dx = ysqrt(1+x^2)dy 
You need to divide by sqrt(1 + y^2) both sides, such that:
xdx = (ysqrt(1+x^2)dy)/(sqrt(1 + y^2))
You need to divide by sqrt(1 + x^2) both sides, such that:
(xdx)/(sqrt(1 + x^2)) = (ydy)/(sqrt(1 + y^2)) 
Integrating both sides yields:
int (xdx)/(sqrt(1 + x^2)) = int (ydy)/(sqrt(1 + y^2)) 
You should come up with the following substitutions, such that:
1 + x^2 = t => 2xdx = dt => xdx = (dt)/2 
1 + y^2 = u => 2ydy = du => ydy = (du)/2 
Changing the variables, yields:
int ((dt)/2)/(sqrt t) = int ((du)/2)/(sqrt u) 
(1/2)int (dt)/(sqrt t) = (1/2) int (du)/(sqrt u) 
(sqrt t)/(1/2) = (sqrt u)/(1/2) => sqrt t = sqrt u => t = u 
Replacing back 1 + x^2 for t and 1 + y^2 for u yields:
1 + x^2 = 1 + y^2 => x^2 = y^2 => y = sqrt(x^2) + c => y = |x| + c 
Hence, evaluating the solution to the given differential equation, using separation of variables, yields the general solution y = |x| + c.

Math

x`sqrt(1+y^2)dx-ysqrt(1+x^2)dy=0` solve using variable separable.

Expert Answers info

Unlock This Answer Now

Related Questions

Ask a Question

Popular Questions