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. $noneofthese$

Answer 1

The correct option is A $x=A_1{y}^2+A_{2}y+A_3$

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