form the differential equations of the family of curves where parameters are given

i) xy = ae^x + be^-x ; ( a, b)

ii) y = (a + bx)e^{k x} ; ( a ; b)

Question

form the differential equations of the family of curves where
parameters are given

i) xy = ae​x + be-x ; ( a, b)

ii) y = (a + bx)ek x ; ( a ; b)

Rajat Sethi · Accepted Answer

1 xy=aex+be-x (i)Differentiating(i) wrt x1
y +xdydx=aex-be-x⇒y +xdydx=aex-be-x
(ii)Again Differentiating (ii) wrt xdydx+1
dydx+xd2ydx2=aex+be-xdydx+dydx+xd2ydx2=xy ( From (i))xd2ydx2+2dydx-xy=0,which is the reqd
 differential equation 2  y=(a+bx)ekx
(i)Differentiating(i) wrt xdydx=(b
1)ekx+k(a+bx)ekxdydx=bekx+ky
(ii)       b=(dydx-ky)e-kxAgain Differentiating(ii) wrt xd2ydx2=kbekx+kdydxd2ydx2=k(dydx-ky)e-kx×ekx +kdydx⇒d2ydx2-2kdydx+k2y=0,which is the reqd
 differential equation

form the differential equations of the family of curves where parameters are given i) xy = ae​x + be-x ; ( a, b) ii) y = (a + bx)ek x ; ( a ; b)

form the differential equations of the family of curves where parameters are given

i) xy = ae^x + be^-x ; ( a, b)

ii) y = (a + bx)e^{k x} ; ( a ; b)