Solve the differential equation:
dy/dx= (x^2+y^2)/(2xy)
I know how to solve this type of problem, but I am struggling getting all of the x's and y's on different sides of the equation. Thank you for your help.
Note that the right-hand side is homogeneous (i.e. the total of exponents of x and y equals 2 for all terms), so a substitution of u=y/x will render the equation separable.
u=y/x
udx+xdu=dy
dy/dx = u+x*du/dx
...
Post if you have difficulties.
To solve the given differential equation, dy/dx = (x^2 + y^2)/(2xy), you can start by multiplying both sides by dx to separate the variables. This will allow us to integrate each side separately.
dy = ((x^2 + y^2)/(2xy))dx
Now, let's rearrange the equation by bringing y^2 and 2xy to the left side and dx to the right side:
(2xy)dy - y^2dx = x^2dx
Next, we can divide both sides of the equation by x^2 to simplify:
(2y/x)dxy - (y^2/x^2)dx = dx
Now, the left side has dy/xdx, which suggests using a substitution. Let's substitute u = y/x. To find du, we need to differentiate u with respect to x:
du = (1/x)dy - (y/x^2)dx
Now, let's substitute this expression for (1/x)dy in the differential equation:
(2u)dx - (y^2/x^2)dx = dx
(2u - y^2/x^2)dx = dx
Now, by comparing the coefficients of dx on both sides, we can write:
(2u - y^2/x^2) = 1
Multiplying both sides by x^2 gives:
2ux^2 - y^2 = x^2
Rearranging the equation, we get:
2ux^2 = y^2 + x^2
Now, substituting u = y/x:
2yx = y^2 + x^2
Rearranging the equation gives:
y^2 - 2yx + x^2 = 0
This is a quadratic equation in terms of y. We can solve it using the quadratic formula:
y = (2x ± √(4x^2 - 4x^2))/2
y = (2x ± 0)/2
y = x
Thus, the solution to the differential equation is y = x.