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Solve This First Order Ordinary Differential Equation Please by Chiboyswag: 5:05pm On Jun 05, 2023 |
(X^2 + y)dx + (x + e^y)dy = 0 1 Share
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Re: Solve This First Order Ordinary Differential Equation Please by nnamdi640: 5:36pm On Jun 05, 2023 |
Pay and I will do it for you |
Re: Solve This First Order Ordinary Differential Equation Please by Westerhoffe(m): 5:42pm On Jun 05, 2023 |
According to ChatGPT: Solve First Order ODE Solve This First Order Ordinary Differential Equation (X^2 + y)dx + (x + e^y)DT To solve the first-order ordinary differential equation (ODE): (X^2 + y)dx + (x + e^y)dy = 0 We can check if it is exact by verifying if the partial derivatives with respect to x and y are equal. Let's calculate those: ∂/∂y (X^2 + y) = 1 ∂/∂x (x + e^y) = 1 Since the partial derivatives are equal, the equation is exact. To solve the exact ODE, we need to find a potential function Φ(x, y) such that: ∂Φ/∂x = X^2 + y ∂Φ/∂y = x + e^y Integrating the first equation with respect to x: Φ(x, y) = (1/3)x^3 + xy + f(y) Here, f(y) is an arbitrary function of y alone. Next, we differentiate Φ(x, y) with respect to y and set it equal to the second equation: ∂Φ/∂y = x + e^y d/dy [(1/3)x^3 + xy + f(y)] = x + e^y Differentiating the terms with respect to y: d/dy (1/3)x^3 + d/dy (xy) + d/dy [f(y)] = x + e^y 0 + x + f'(y) = x + e^y We can see that f'(y) = e^y. Integrating both sides with respect to y: ∫f'(y) dy = ∫e^y dy f(y) = e^y + C Here, C is the constant of integration. Substituting this value of f(y) into our potential function: Φ(x, y) = (1/3)x^3 + xy + e^y + C Now, we have found the potential function Φ(x, y). To find the solution to the exact ODE, we equate Φ(x, y) to a constant, K: (1/3)x^3 + xy + e^y + C = K This is the general solution to the given first-order ODE. |
Re: Solve This First Order Ordinary Differential Equation Please by Chiboyswag: 6:02pm On Jun 05, 2023 |
Thanks brother 🥰 |
Re: Solve This First Order Ordinary Differential Equation Please by clockwisereport: 6:36pm On Jun 08, 2023 |
@chiboyswag 1 Like
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Re: Solve This First Order Ordinary Differential Equation Please by Chiboyswag: 2:24am On Jun 13, 2023 |
clockwisereport:thank you Sir |
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