solve-
dy/dx = 2xy + (y^2/x^2)
Looks like a Riccati equation. Standard methods, such as those at
http://www.sosmath.com/diffeq/first/riccati/riccati.html
will lead to the solution
y = xe^(x^2) / (cx + e^(x^2) - πix erf(ix))
where erf(x) is the error function.
Use wolframalpha dot com - it will solve yr equation and show all steps.
To solve the differential equation dy/dx = 2xy + (y^2/x^2), we can use the method of separable variables. This method involves separating the variables and integrating both sides of the equation with respect to their respective variables.
Here's how to solve the differential equation step-by-step:
Step 1: Separate the variables
Start by moving all the terms with y to one side of the equation and all the terms with x to the other side. You can do this by subtracting 2xy from both sides and multiplying both sides by x^2:
x^2 * dy/dx - 2xy = y^2
Step 2: Rearrange the equation
Rearrange the equation to group the y terms on one side and the x terms on the other side:
x^2 * dy/dx - 2xy - y^2 = 0
Step 3: Factor out y
Factor out y from the equation:
y * (x^2 * dy/dx - 2x - y) = 0
Step 4: Set each factor equal to zero
Since the product of two factors is zero, we can set each factor equal to zero and solve for y separately:
y = 0
and
x^2 * dy/dx - 2x - y = 0
Step 5: Solve for y = 0
The first factor, y = 0, gives us one solution.
Step 6: Solve for x^2 * dy/dx - 2x - y = 0
To solve the second factor, x^2 * dy/dx - 2x - y = 0, we can use an auxiliary equation to transform it into a linear first-order differential equation. Let's substitute u = y/x:
dy/dx = x * du/dx + u
Now, substitute this into the original equation:
x^2 * (x * du/dx + u) - 2x - u = 0
Simplify the equation:
x^3 * du/dx + x^2 * u - 2x - u = 0
Combine like terms:
x^3 * du/dx + (x^2 - 1) * u - 2x = 0
Step 7: Solve the linear first-order differential equation
The resulting equation, x^3 * du/dx + (x^2 - 1) * u - 2x = 0, is a linear first-order differential equation in the form of du/dx + P(x) * u = Q(x), where P(x) = (x^2 - 1) and Q(x) = 2x.
To solve this linear differential equation, we can use an integrating factor.
The integrating factor is given by the formula:
IF(x) = e^(integral[P(x)dx])
In this case, the integrating factor becomes:
IF(x) = e^(integral[(x^2 - 1)dx])
= e^(x^3/3 - x)
Multiply both sides of the linear differential equation by the integrating factor IF(x) and simplify:
e^(x^3/3 - x) * [x^3 * du/dx + (x^2 - 1) * u - 2x] = 0
Step 8: Integrate both sides of the equation
Integrate both sides of the equation to solve for u:
∫ e^(x^3/3 - x) * [x^3 * du/dx + (x^2 - 1) * u - 2x] dx = ∫ 0 dx
After integrating, you should end up with an expression for u(x). You can then substitute u back into the equation u = y/x to get the solution y(x).
Note: Due to the complexity of the integrals involved, the final solution may not have a closed-form expression and may require numerical methods.