(a). \( \frac{d y}{d x}+2 x y^{2}=0 \) (b). \( \frac{d y}{d x}=e^{3 x+2 y} \)
Solución de inteligencia artificial de Upstudy
Responder
Solución

¡Inicia sesión para desbloquear respuestas gratis!
Una plataforma de aprendizaje en la que confían millones de estudiantes y profesores reales.

Respondido por UpStudy AI y revisado por un tutor profesional
The Deep Dive
To solve the first differential equation, we identify it as a separable equation: \( \frac{dy}{dx} = -2xy^2 \). We can separate variables by rearranging it to \( \frac{dy}{y^2} = -2x \, dx \). Integrating both sides will yield a solution that involves the logarithm or another algebraic manipulation, potentially leading to a family of curves defined in terms of constants. For the second equation \( \frac{dy}{dx} = e^{3x + 2y} \), this is also separable. Rearranging gives us \( \frac{dy}{e^{2y}} = e^{3x} \, dx \). This allows us to perform an integration on both sides, giving rise to an exponential function in \( y \). The solution will also involve a constant of integration, which can later be used in initial conditions if provided.
preguntas relacionadas
