Question

Resolva a equação diferencial separável dy/dx=y√X que satisfaz a condição inicial y(9)=3

Answer 1

Olá


$\displaystyle \mathsf{ \frac{dy}{dx}=y \sqrt{x} }\\\\\\\mathsf{ \frac{dy}{y}= \sqrt{x} dx }$

Integrando dos dois lados

$\displaystyle \mathsf{ \int\frac{dy}{y}= \int\sqrt{x} dx }\\\\\\\mathsf{\ell n|y|= \frac{2 \sqrt{x^3} }{3}~+~C }$

Dados do enunciado
y(9) = 3

x = 9 ;  y = 3

Substituindo na EDO

$\displaystyle \mathsf{\ell n|3|= \frac{2 \sqrt{9^3} }{3}~+~C }\\\\\\\mathsf{\ell n|3|= \frac{2\cdot 27 }{3}~+~C }\\\\\\\mathsf{\ell n|3|=18~+~C }\\\\\\\boxed{\mathsf{C=\ell n |3|-18}}$

$\displaystyle \mathsf{\ell n|y|= \frac{2 \sqrt{x^3} }{3}~+~\ell n(3)-18 }$


Deixando o y de forma explícita.

Aplica-se exponencial em ambos os lados, para cancelarmos o ln


$\displaystyle \mathsf{e^{\ell n|y|}=e^{ \frac{2 \sqrt{x^3} }{3}~+~\ell n(3)-18 }}\\\\\\\boxed{\boxed{\mathsf{y=3e^{ \frac{ 2\sqrt{x^3} }{3} -18}}}}$