미분방정식 전체 문제

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.)

In your own words, what is an initial value problem, and how is it different than a differential equation?

In your own words, describe what it means for a function to be a solution to a differential equation.

How can we verify that a function is a solution to a differential equation?

Describe the difference between a particular solution and a general solution.

Why might we use a graphical or numerical technique to study solutions to a differential equation instead of simply solving the differential equation to find an analytic solution?