Question

The solution of the differential equation $y_1{y}_3=3y_2^2$ is

• A. $x=A_1{y}^2+A_{2}y+A_3$
• B. $x=A_{1}y+A_2$
• C. $x=A_1{y}^2+A_{2}y$
• D. $y=A_{1}x+A_2$

Answer 1

Answer: (A)

$y_1{y}_3=3y_2^2$
$\frac{y_3}{y_2}=3\frac{y_2}{y_1}\Rightarrow \ell n{y}_2=3\ell n{y}_1+\ell nc$
$y_2=c{y}_1{}^3$
$\frac{y_2}{y_1^2}=c{y}_1$
$-\frac{1}{y_1}=cy+c_2$
$\frac{dx}{dy}=-cy-c_2$
$x=-\frac{c{y}_2}{2}-c_{2}y+c_3$
$\therefore x=A_1{y}_1+A_{2}y+A_3$