solve the separable differential equation. (use c for any needed constant.)\n\\( \\frac { d y } { d x } =…
solve the separable differential equation. (use c for any needed constant.)\n\\( \\frac { d y } { d x } = \\frac { x + 7 } { y - 7 } \\)
Answer
Explanation:
Step1: Separate variables
$$(y - 7)dy=(x + 7)dx$$
Step2: Integrate both sides
$$\int(y - 7)dy=\int(x + 7)dx$$ $$\frac{y^{2}}{2}-7y=\frac{x^{2}}{2}+7x + C$$
Step3: Simplify the equation
Multiply through by (2) to get (y^{2}-14y=x^{2}+14x + 2C). Let (C_{1}=2C), then (y^{2}-14y-(x^{2}+14x)=C_{1}).
Answer:
(y^{2}-14y-(x^{2}+14x)=C) (where (C) is an arbitrary constant)