find the general solution
x^2-2(e^y)dy/dx = y^3+e(x^2)
i am getting the y^3 dx when i do seperation of variables how do i deal with this.
To find the general solution for the given equation, we will use the method of separation of variables. Let's go step by step:
1. Start with the given differential equation: x^2 - 2(e^y)dy/dx = y^3 + e(x^2).
2. Rearrange the equation by moving all terms involving x to one side and all terms involving y to the other side: x^2 - y^3 = 2(e^y)dy/dx - e(x^2).
3. Now, separate the variables. Move all terms involving y and dy to one side, and all terms involving x and dx to the other side: (x^2 - y^3)dx = 2(e^y)dy - e(x^2)dx.
4. Divide both sides of the equation by (x^2 - y^3) to isolate the dx and dy terms: dx = (2(e^y)dy - e(x^2)dx) / (x^2 - y^3).
5. Rearrange the equation to group the dx and dy terms separately: dx + e(x^2)dx = 2(e^y)dy - e(x^2)dx.
6. Notice that terms involving dx appear on both sides of the equation. To simplify, add e(x^2)dx to both sides: dx/(x^2) = 2(e^y)dy/(x^2).
7. Now, integrate both sides of the equation. Integrate dx/(x^2) with respect to x and 2(e^y)dy/(x^2) with respect to y: ∫(dx/(x^2)) = ∫(2(e^y)dy/(x^2)).
8. Integrating both sides yields: -1/x = 2e^y/x + C, where C is the constant of integration.
9. To simplify further, multiply both sides by -x: 1 = -2e^y + Cx.
10. Finally, rearrange the equation to isolate y: -2e^y = 1 - Cx. Divide by -2 and take the natural logarithm of both sides: y = ln((1 - Cx)/(-2)).
Therefore, the general solution to the given differential equation is y = ln((1 - Cx)/(-2)), where C is the constant of integration.