solve the differential equation (x^2-xy+y^2)dy-xydx=0
plz show me working
The last time I did differential equations was 55 years ago, perhaps one of our other math tutors can kick in here
http://www.wolframalpha.com/input/?i=solve+x+y+dx+%3D+(x%5E2+-x+y+%2By%5E2)dy
i need the step
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But, note that
y' = xy/(x^2-xy+y^2)
This is homogeneous, so let y=vx
Then y' = v + xv'
v + xv' = vx^2/(x^2-vx+x^2v^2)
v + xv' = v/(v^2-v+1)
xv' = (v^2-2v+1)/(v^2-v+1)
(v^2-v+1)/(v-1)^2 dv = -1/x dx
Now just do a long division and partial fractions, and integrate both sides. Then substitute back using v = y/x.
To solve the given differential equation:
Step 1: Rearrange the equation to separate the variables.
Start by moving the term involving dy to the other side of the equation:
(x^2 - xy + y^2) dy = xy dx
Step 2: Divide both sides by the coefficient of dy.
Divide both sides of the equation by (x^2 - xy + y^2):
dy/dx = (xy) / (x^2 - xy + y^2)
Step 3: Rewrite the equation using the notation y' for dy/dx:
Replace dy/dx with y' to rewrite the equation as:
y' = (xy) / (x^2 - xy + y^2)
Step 4: Solve the equation by separating variables.
Rearrange the equation by multiplying both sides by (x^2 - xy + y^2):
(x^2 - xy + y^2) y' = xy
Step 5: Simplify the equation.
Expand the left-hand side of the equation:
x^2y' - xy' + y^2y' = xy
Step 6: Group the terms involving y' and factor.
Factor out y' from the left-hand side:
y' (x^2 - x + y^2) = xy
Step 7: Divide both sides by (x^2 - x + y^2) to solve for y':
Divide both sides of the equation by (x^2 - x + y^2):
y' = (xy) / (x^2 - x + y^2)
Step 8: Integrate both sides to find y(x).
Integrate with respect to x on both sides:
∫ y' dx = ∫ (xy) / (x^2 - x + y^2) dx
Step 9: Solve the integral on both sides.
The integral on the left-hand side can be solved as y:
y = ∫ (xy) / (x^2 - x + y^2) dx
The integral on the right-hand side may not have a simple analytical solution in terms of elementary functions. In this case, you may need to use numerical methods or approximation techniques to find an approximate solution.
Therefore, the differential equation (x^2 - xy + y^2) dy - xy dx = 0 is simplified to y = ∫ (xy) / (x^2 - x + y^2) dx.