Question

The differential equation $\frac{d^{2}y}{d{x}^2}+16y=0$ for $y\left(x\right)$ with the two boundary conditions ${\begin{array}{c}\frac{dy}{dx}\end{array}|}_{x=0}=1$ and ${\begin{array}{c}\frac{dy}{dx}\end{array}|}_{x=\frac{1}{2}}=-1$ has

• A. no solution
• B. exactly two solutions
• C. exactly one solution
• D. infinitely many solutions

Answer 1

The correct option is B exactly two solutions

$\frac{d^{2}y}{d{x}^2}+16y=0$
$(D^2+16)y=0$
Let $D^2=m^2$
$m^2+16=0$ (this is a complex equation)
$m=\pm 4i=0\pm 4i$
$y=(C_{}cos4x+C_{2}sin4x)e^{ox}$
$\Rightarrow y=C_{1}cos4x+C_{2}sin4x$
$\Rightarrow y^{\prime }=-4C_{1}sin4x+4C_{2}cos4x$
$y^{\prime }\left(0\right)=4C_2=1$
$C_2=\frac{1}{4}$
$y^{\prime }\frac{\pi }{2}=-1$
$-1=-4C_{1}sin2\pi +4C_{2}cos2\pi$
$-1=4C_2$
$C_2=-\frac{1}{4}$
So, we are not sure about $C_2$. Hence no solution.