Question

A solução da equação de variáveis separáveis 4 - xy + dy/dx= x - 4y, com y(0)=3, é:

Answer 1

$4 - xy + \frac{dy}{dx}= x - 4y\\\\ \frac{dy}{dx}= x-4y+xy-4 \\\\ \frac{dy}{dx}= x+xy -4-4y\\\\ \frac{dy}{dx}= x(1+y) -4(1+y)\\\\ \frac{dy}{dx} = (1+y)(x-4)\\\\ \frac{dy}{1+y} = (x-4)dx\\\\ \int \frac{dy}{1+y} = \int(x-4)dx\\\\\ ln(1+y)= \frac{x^2}{2} -4x+C\\\ 1+y = e^{\frac{x^2}{2} -4x+C}\\\\ 1+y = e^{\frac{x^2}{2}-4x}*e^C\\\\ 1+y= K*e^{\frac{x^2}{2}-4x}\\\\\ \boxed{\boxed{y(x)=K*e^{\frac{x^2}{2}-4x} - 1}}\\\\ y(0)= K*e^{ 0 }-1 = 3 \\\\ K-1=3\\K=4\\\\ \boxed{\boxed{y(x)=4e^{\frac{x^2}{2}-4x} - 1}}$