Question

calcule equações diferenciais de dy/dx =3 y^(2/3), y(0) = 0

Answer 1

Olá



$\displaystyle\mathsf{ \frac{dy}{dx}~=~3y^{ \frac{2}{3}}}\\\\\\\\\mathsf{ \frac{dy}{3y^{ \frac{2}{3} }}~=~dx}\\\\\\\text{Integrando dos dois lados}\\\\\\\\\mathsf{ \int \frac{dy}{3y^{ \frac{2}{3} }}~=\int dx}$

$\displaystyle \text{Resolvendo a primeira integral separadamente}\\\\\\\\\mathsf{ \int \frac{dy}{3y^{ \frac{2}{3} }}~= ~\frac{1}{3} ~\int y^{\frac{-2}{3} }dy~=~\frac{1}{3}~ \cdot \frac{y^{- \frac{2}{3}+1 }}{- \frac{2}{3}+1}~=~\frac{1}{\diagup\!\!\!\!3}\cdot \diagup\!\!\!\!3y^{ \frac{1}{3} } ~=~\boxed{\mathsf{y^{ \frac{1}{3} }}}}$


$\displaystyle\mathsf{y^{ \frac{1}{3} }~=~x+c}\\\\\\\mathsf{ \sqrt[3]{y}~= ~x+c }\\\\\\\\\\\text{Condicao inicial}\\\\ \mathsf{y(0) = 0}\\\\\\\mathsf{ \sqrt[3]{0}~=~0+c }\\\\\boxed{\mathsf{c=0}}\\\\\\\mathsf{ \sqrt[3]{y}~=~x }\\\\\\\text{Elevando os dois lados ao cubo}\\\\\\\boxed{\mathsf{y= x^3} }}\qquad\qquad\Longleftarrow\qquad\text{Resposta final}$