Solve the boundary-value problem, if possible. $y ^ { t t } + 14 y ^ { t } + 49 y = 0 , y ( 0 ) = 0 , y ( 1 ) = 3$
A) $y ( x ) = 7 x e ^ { ( - 3 x + 3 ) }$
B) $y ( x ) = 3 e ^ { ( - 7 x - 7 ) }$
C) $y ( x ) = 3 x e ^ { ( - 7 x - 7 ) }$
D) $y ( x ) = 3 x e ^ { ( - 7 x + 7 ) }$
E) No solution

Question

Quizplus · Accepted Answer

The characteristic equation for the given differential equation is 
$$r^{2}+14r+49=(r+7)^{2}=0$$
Therefore, the general solution is 
$$y(x)=c_1e^{-7x}+c_2xe^{-7x}$$
Using the given boundary conditions, we can find the values of $c_1$ and $c_2$ as: 
$$y(0)=c_1=0$$
$$y(1)=c_2e^{-7}+0=3 \implies c_2=3e^7$$
Thus, the solution to the given boundary value problem is 
$$y(x)=3xe^{-7x+7}$$
which matches with option D.

Solve the Boundary-Value Problem, If Possible ytt+14yt+49y=0,y(0)=0,y(1)=3y ^ { t t } + 14 y ^ { t } + 49 y = 0 , y ( 0 ) = 0 , y ( 1 ) = 3ytt+14yt+49y=0,y(0)=0,y(1)=3

Solve the Boundary-Value Problem, If Possible $y ^ { t t } + 14 y ^ { t } + 49 y = 0 , y ( 0 ) = 0 , y ( 1 ) = 3$