solve for dy/dx=(x^2+y^2)/xy by substitution.
Plz help step
http://www.mathsisfun.com/calculus/differential-equations-homogeneous.html
LOOK at the example at the end :)
To solve the given differential equation dy/dx = (x^2 + y^2)/(xy) using substitution, we can let v = y/x. This will help us simplify the equation and make it easier to solve.
Here are the steps to solve the equation:
Step 1: Find the derivative of v with respect to x using the quotient rule.
dv/dx = (x(dy/dx) - y)/(x^2)
Step 2: Substitute dv/dx and y/x into the original equation.
(x^2 + y^2)/(xy) = (x(dy/dx) - y)/(x^2)
Step 3: Simplify the equation by cross-multiplying and rearranging the terms:
x^2 + y^2 = xy(dy/dx) - yx
Step 4: Rearrange the equation to isolate dy/dx on one side:
xy(dy/dx) = x^2 + y^2 - yx
Step 5: Simplify the equation further:
dy/dx = (x^2 + y^2 - yx)/(xy)
Step 6: Factor out the common term (x - y) from the numerator:
dy/dx = (x(x - y) + y(x - y))/(xy)
Step 7: Cancel out the common factor of (x - y) in the numerator and denominator:
dy/dx = (x + y)/x
So, the solution to the differential equation dy/dx = (x^2 + y^2)/(xy) after substitution is dy/dx = (x + y)/x.