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Verify that the indicated family of functions is a solution of the given differential equation. Assume an appropriate interval $I$ of definition for each solution. $dx_{2}d_{2}y??8dxdy?+16y=0;y=c_{1}e_{4x}+c_{2}xe_{4x}$ When $y=c_{1}e_{4x}+c_{2}xe_{4x}$, $dxdy?dx_{2}d_{2}y??==?$ Thus, in terms of $x$, $dx_{2}d_{2}y??8dxdy?+16y=+16(c_{1}e_{4x}+c_{2}xe_{4x})$
Verify that the indicated family of functions is a solution of the given differential equation. Assume an appropriate interval $I$ of definition for each solution. $dtdP?=P(1?P);P=1+c_{1}e_{t}c_{1}e_{t}?$ When $P=1+c_{1}e_{t}c_{1}e_{t}?$ $dtdP?=$ Thus, in terms of $t$, $dtdP??1+c_{1}e_{t}c_{1}e_{t}?(1?1+c_{1}e_{t}c_{1}e_{t}?)?=?1+c_{1}e_{t}c_{1}e_{t}?(1?1+c_{1}e_{t}c_{1}e_{t}?)=?$

d2ydx2?8dydx+16y=0so we have solution y=c1e4x+c2xe4xso diffrentiate it with respect to xdydx=4c1e4x