Question

Determine a família de soluções da equação diferencial separável- EDO

Answer 1

Resposta:

$y=sen\left(\frac{x^2}{2}+c\right)$

Explicação passo a passo:

$\frac{dy}{dx}=x\sqrt{1-y^2}\\\\\\\frac{dy}{\sqrt{1-y^2}}=xdx\\\\\\\:\int \:\frac{dy}{\sqrt{1-y^2}}=\int \:xdx\\\\\\Pela\:tabela\:das\:EDO's\:temos\:que:\\\\\\\int \frac{dx}{\sqrt{a^2-x^2}\:}=arcsen\left(\frac{x}{a}\right)+C\\\\\\Se,\:\:\int \frac{dy}{\sqrt{1-y^2}}=\int \frac{dy}{\sqrt{1^2-y^2}}\\\\\\Entao:\\\\\\arcsen\left(y\right)=\frac{x^2}{2}+C\\\\\\Sabendo\:que,\:f\left(a\right)=b\:e\:f'\left(b\right)=a\\\\\\sen\left(\frac{x^2}{2}+C\right)=y\\\\\\y=sen\left(\frac{x^2}{2}+C\right)$