Final answer:
To solve the given differential equation (2x²-y²) dx+ (xy-x³y-1) dy = 0 using the method for solving homogeneous equations, we can convert it to a separable equation by letting u = y/x. After solving the resulting equation and simplifying, we obtain the implicit solution F(x, y) = y² - xy - Cx^2e^x, where C is an arbitrary constant.
Step-by-step explanation:
To solve the given differential equation (2x²-y²) dx+ (xy-x³y-1) dy = 0, we can use the method for solving homogeneous equations. Let's rewrite the equation in the form dx/x- dy/y = (x²-xy³+y²-1)/(xy-x³y) = 0. Now, let u = y/x and differentiate with respect to x, du/dx = (dy/dx - u) / x. Substituting this into the equation and simplifying, we get du/(u-u²) = dx/x.
Now, we can solve the resulting separable equation. Integrating both sides, we have ln|u-u²| = ln|x| + C, where C is the constant of integration. Exponentiating both sides, we get u-u² = Ce^x. Rearranging the equation and substituting u = y/x, we have y/x - (y/x)² = Ce^x. Simplifying, we get y² - xy = Cx^2e^x.
Therefore, the implicit solution to the given differential equation is F(x, y) = y² - xy - Cx^2e^x, where C is an arbitrary constant.