Question

- Resolva as seguintes equações diferenciais, por separação de variáveis.

Answer 1

$\Large\boxed{\begin{array}{l}\tt a)~\sf\dfrac{dy}{dx}=\dfrac{10}{x^2+1},~y(0)=0\\\\\sf dy=\dfrac{10dx}{x^2+1}\\\\\displaystyle\sf\int dy=10\int\dfrac{dx}{x^2+1}\\\sf y(x)=10 arctg(x)+c\\\sf y(0)=10 arctg(0)+c\\\sf\\\sf c=0\\\sf y(x)=10arctg(x)\end{array}}$

$\Large\boxed{\begin{array}{l}\tt b)~\sf\dfrac{dy}{dx}=\dfrac{y^3}{x^2},y(4)=9\\\\\sf\dfrac{dy}{y^3}=\dfrac{dx}{x^2}\\\\\displaystyle\sf\int\dfrac{dy}{y^3}=\int\dfrac{dx}{x^2}\\\\\sf-\dfrac{1}{2y^2}=-\dfrac{1}{x}+c\\\sf\dfrac{1}{2y^2}=\dfrac{1}{x}+c\\\sf 2y^2=x+c\\\sf y^2=\dfrac{1}{2}x+c\\\sf y(x)=\sqrt{\dfrac{1}{2}x+c}\\\sf y(4)=\sqrt{\dfrac{1}{2}\cdot 4+c}\\\sf\sqrt{c+2}=9\\\sf(\sqrt{c+2})^2=9^2\\\sf c+2=81\\\sf c=81-2\\\sf c=79\\\sf y(x)=\sqrt{\dfrac{1}{2}x+79}\end{array}}$