# form the differential equation for the following family of curves y=c1*(e^x)+(e^c1)*e^(2x)+(ln(c1))*e^(3x)

Jitender Singh IIT Delhi
8 years ago
Ans:
$y = c_{1}e^{x} + e^{c_{1}}e^{2x} + lnc_{1}.e^{3x}$
$y' = c_{1}e^{x} + 2e^{c_{1}}e^{2x} + 3 lnc_{1}.e^{3x}$
$y'' = c_{1}e^{x} + 4e^{c_{1}}e^{2x} + 9lnc_{1}.e^{3x}$
$y''' = c_{1}e^{x} + 8e^{c_{1}}e^{2x} + 27lnc_{1}.e^{3x}$
Differential equation of family of curves:
You can solve this system of equations by matrix method. Matrix would be of 4 columns & 4 rows.
First column elements will be 1, 1, 1 & 1, second column elements will be 1, 2, 4 & 8, third column will be 1, 3, 9 & 27 ans 4thcolumn will be y, y’, y’’ & y’’’.
Detrminent of matrix is zero.
Thanks & Regards
Jitender Singh
IIT Delhi