Question

Resolva a equação separável y'=(4x² - senx), y(pi) = 1.

Answer 1

Resposta:

$y(x)=\dfrac{4}{3}x^3+\cos{x}+\bigg(2-\dfrac{4}{3}\pi^3\bigg)$

Explicação passo-a-passo:

$\dfrac{dy}{dx} =(4x^2-\sin{x}), \ y(\pi)=1 \\\\ dy=(4x^2-\sin{x})\, dx \ \to \ y = \int {(4x^2-\sin{x})} \, dx \ \to \\\\ \to \ y = 4\int {x^2}\, dx+\int {(-\sin{x})}\,dx \ \to\ y=4\bigg{(}\dfrac{x^3}{3}\bigg{)}+C_{1}+\cos{x}+C_{2}\ \to \\\\ \to\ y(x)=\dfrac{4}{3}x^3+\cos{x}+C\\\\ \text{Portanto:}\\\\ y(\pi)=1\ \to\ \dfrac{4}{3}\pi^3+\cos{\pi}+C=1\ \to\ C=1-(-1)-\dfrac{4}{3}\pi^3\ \to\\\\ \to\ C=2-\dfrac{4}{3}\pi^3\\\\ \text{Logo:}\\\\ y(x)=\dfrac{4}{3}x^3+\cos{x}+\bigg(2-\dfrac{4}{3}\pi^3\bigg)$