Question

The differential equation which represents the family of curves $y=c_1{e}^{c_{2}x}$, where $c_1$ and $c_2$ are arbitrary constants, is

• A. $y{}^{\prime \prime }=y^{\prime }y$
• B. $yy{}^{\prime \prime }=(y^{\prime }{)}^2$
• C. $yy{}^{\prime \prime }=y^{\prime }$
• D. $y^{\prime }=y^2$

Answer 1

The correct option is B $yy{}^{\prime \prime }=(y^{\prime }{)}^2$

$y=C_1{e}^{C_{2}x}$

$y_1=C_1{C}_2{e}^{C_{2}x}$

$y_2=C_1{C}_2^2{e}^{C_{2}x}$

we can say that

$y{y}_2=(y_1{)}^2$

that is $y{y}^{\prime \prime }=(y^{{}^{\prime }}{)}^2$