Chapter 7

Find a power series solution for the following differential equations. $$ (x-7) y^{\prime}+2 y=0 $$

Find a power series solution for the following differential equations. $$ \left(1+x^{2}\right) y^{\prime \prime}-4 x y^{\prime}+6 y=0 $$

Find a power series solution for the following differential equations. $$ x^{2} y^{\prime \prime}-x y^{\prime}-3 y=0 $$

Find a power series solution for the following differential equations. $$ y^{\prime \prime}-8 y^{\prime}=0, \quad y(0)=-2, \quad y^{\prime}(0)=10 $$

Find a power series solution for the following differential equations. $$ y^{\prime \prime}-2 x y=0, \quad y(0)=1, \quad y^{\prime}(0)=-3 $$

Find a power series solution for the following differential equations. \(\begin{array}{llll}116 . & \text { The } & \text { differential } & \text { equation }\end{array}\) \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-1\right) y=0\) is a Bessel equation of order 1. Use a power series of the form \(y=\sum_{n=0}^{\infty} a_{n} x^{n}\) to find the solution.

True or False? Justify your answer with a proof or a counterexample. If \(y\) and \(z\) are both solutions to \(y^{n}+2 y^{\prime}+y=0\), then \(y+z\) is also a solution.

True or False? Justify your answer with a proof or a counterexample. The following system of algebraic equations has a unique solution: $$ \begin{array}{l} 6 z_{1}+3 z_{2}=8 \\ 4 z_{1}+2 z_{2}=4 \end{array} $$

True or False? Justify your answer with a proof or a counterexample. \(y=e^{x} \cos (3 x)+e^{x} \sin (2 x)\) is a solution to the second-order differential equation \(y^{\prime \prime}+2 y^{\prime}+10=0\).

True or False? Justify your answer with a proof or a counterexample. To find the particular solution to a second-order differential equation, you need one initial condition.

Classify the differential equation. Determine the order, whether it is linear and, if linear, whether the differential equation is homogeneous or non homogeneous. If the equation is second-order homogeneous and linear, find the characteristic equation. $$ y^{\prime \prime}-2 y=0 $$

Classify the differential equation. Determine the order, whether it is linear and, if linear, whether the differential equation is homogeneous or non homogeneous. If the equation is second-order homogeneous and linear, find the characteristic equation. $$ y^{\prime \prime}-3 y+2 y=\cos (t) $$

Classify the differential equation. Determine the order, whether it is linear and, if linear, whether the differential equation is homogeneous or non homogeneous. If the equation is second-order homogeneous and linear, find the characteristic equation. $$ \left(\frac{d y}{d t}\right)^{2}+y y^{\prime}=1 $$

Classify the differential equation. Determine the order, whether it is linear and, if linear, whether the differential equation is homogeneous or non homogeneous. If the equation is second-order homogeneous and linear, find the characteristic equation. $$ \frac{d^{2} y}{d t^{2}}+t \frac{d y}{d t}+\sin ^{2}(t) y=e^{t} $$

For the following problems, find the general solution. $$ y^{n}+9 y=0 $$

For the following problems, find the general solution. $$ y^{\prime \prime}+2 y^{\prime}+y=0 $$

For the following problems, find the general solution. $$ y^{\prime \prime}-x^{2}=-3 y^{\prime}-\frac{9}{4} y+3 x $$

For the following problems, find the solution to the initial value problem, if possible. $$ y^{\prime \prime}=3 y-\cos (x), \quad y(0)=\frac{9}{4}, \quad y^{\prime}(0)=0 $$

For the following problems, find the solution to the boundary-value problem. $$ 4 y^{\prime}=-6 y+2 y^{\prime \prime}, \quad y(0)=0, \quad y(1)=1 $$

For the following problems, find the solution to the boundary-value problem. $$ y^{\prime \prime}=3 x-y-y^{\prime}, \quad y(0)=-3, \quad y(1)=0 $$

For the following problem, set up and solve the differential equation. 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. The following problems consider the "beats" that occur when the forcing term of a differential equation causes "slow" and "fast" amplitudes. Consider the general differential equation \(a y^{\prime \prime}+b y=\cos (\omega t)\) that govems undamped motion. Assume that \(\sqrt{\frac{b}{a}} \neq \omega .\)

An opera singer is attempting to shatter a glass by singing a particular note. The vibrations of the glass can be modeled by \(y^{\prime \prime}+a y=\cos (b t), \quad\) where \(y^{\prime \prime}+a y=0\) represents the natural frequency of the glass and the singer is forcing the vibrations at \(\cos (b t)\). 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.)