Question

Найти общее решение дифференциального уравнения xy'-2y=2x^4. нужно

Answer 1

Решить $\displaystyle x\frac{\text{d}y(x)}{\text{d}x}-2y(x)=2x^4$ для $\displaystyle y(x).$

$\displaystyle x\frac{\text{d}y(x)}{\text{d}x}-2y(x)=2x^4;$

$\displaystyle \frac{\text{d}y(x)}{\text{d}x}-\frac{2y(x)}{x}=2x^3;$

$\displaystyle \frac{1}{x^2}\frac{\text{d}y(x)}{\text{d}x}-\frac{2}{x^3}y(x)=2x;$

$\displaystyle \frac{1}{x^2}\frac{\text{d}}{\text{d}x}\Big(y(x)\Big)+y(x)\frac{\text{d}}{\text{d}x}\left(\frac{1}{x^2}\right)=2x;$

$\displaystyle \frac{\text{d}}{\text{d}x}\left(\frac{y(x)}{x^2}\right)=2x;$

$\displaystyle \int\frac{\text{d}}{\text{d}x}\left(\frac{y(x)}{x^2}\right)\text{d}x=\int2x\text{d}x;$

$\displaystyle \frac{y(x)}{x^2}=x^2+\text{C};$

$\displaystyle y(x)=x^4+\text{C}x^2;$

$\displaystyle\therefore{y(x)=\boxed{x^2\left(x^2+\text{C}\right)}}\phantom{.}.$