Question

resultado da equação diferencial dy/dx = y/x

Answer 1

Olá


Resolvendo pelo método de separação de variáveis


$\displaystyle \mathsf{ \frac{dy}{dx} = \frac{y}{x} }\\\\\\\mathsf{ \frac{dy}{y} = \frac{dx}{x}}\\\\\\\text{Integra dos dois lados}\\\\\\\mathsf{\int \frac{dy}{y} = \int\frac{dx}{x}}\\\\\\\mathsf{\ell ny~=~\ell n x + c}\\\\\\\text{Aplica exponencial dos dois lados para eliminar o logaritmo natural}\\\\\\\mathsf{e^{\ell ny}~=~e^{\ell n x + c}}\\\\\\\boxed{\mathsf{y=x\cdot c}}$