2계 선형 제차 미분방정식

Classify the differential equation. Determine the order, whether it is linear and, if linear, whether the differential equation is homogeneous or nonhomogeneous. If the equation is second-order homogeneous and linear, find the characteristic equation.

$\frac{{d}^{2}y}{d{t}^{2}}+t\frac{dy}{dt}+{\sin }^{2}(t)y={e}^{t}$

For the following problems, find the general solution.

${y}^{''}+9y=0$

For the following problems, find the general solution.

${y}^{''}+2{y}^{'}+y=0$

For the following problems, find the general solution.

${y}^{''}-2{y}^{'}+10y=4x$

For the following problems, find the general solution.

${y}^{''}=\cos (x)+2{y}^{'}+y$

For the following problems, find the general solution.

${y}^{''}+5y'+4y=x+{e}^{2x}$

For the following problems, find the general solution.

${y}^{''}=3{y}^{'}+x{e}^{-x}$

For the following problems, find the general solution.

${y}^{''}-{x}^{2}=-3{y}^{'}-\frac{9}{4}y+3x$

For the following problems, find the general solution.

${y}^{''}=2\cos x+{y}^{'}-y$

For the following problems, find the solution to the initial-value problem, if possible.

${y}^{''}+4{y}^{'}+6y=0,$ $y(0)=0,$ ${y}^{'}(0)=\sqrt{2}$

For the following problems, find the solution to the initial-value problem, if possible.

${y}^{''}=3y-\cos (x),$ $y(0)=\frac{9}{4},$ ${y}^{'}(0)=0$

For the following problems, find the solution to the boundary-value problem, if possible.

$4{y}^{'}=-6y+2{y}^{''},$ $y(0)=0,$ $y(1)=1$

For the following problems, find the solution to the boundary-value problem, if possible.

${y}^{''}=3x-y-{y}^{'},$ $y(0)=-3,$ $y(1)=0$

The motion of a swinging pendulum for small angles $\theta$ can be approximated by $\frac{{d}^{2}\theta }{d{t}^{2}}+\frac{g}{L}\theta =0,$ where $\theta$ is the angle the pendulum makes with respect to a vertical line, g is the acceleration resulting from gravity, and L is the length of the pendulum. Find the equation describing the angle of the pendulum at time $t,$ assuming an initial displacement of ${\theta }_{0}$ and an initial velocity of zero.

Find the general solution to this equation (Hint: call ${\omega }_{0}=\sqrt{b/a}$).

Assuming the system starts from rest, show that the particular solution can be written as $y=\frac{2}{a({\omega }_{0}{}^{2}-{\omega }^{2})}\sin \left(\frac{{\omega }_{0}-\omega t}{2}\right)\sin \left(\frac{{\omega }_{0}+\omega t}{2}\right).$

[T] Using your solutions derived earlier, plot the solution to the system $2{y}^{''}+9y=\cos (2t)$ over the interval $t=[-50,50].$ Find, analytically, the period of the fast and slow amplitudes.

An opera singer is attempting to shatter a glass by singing a particular note. The vibrations of the glass can be modeled by ${y}^{''}+ay=0,$ where $a>0$ represents the natural frequency of the glass and the singer is forcing the vibrations at $\cos (bt).$ For what value $b$ would the singer be able to break that glass? (Note: in order for the glass to break, the oscillations would need to get higher and higher.)