Three solutions of a second-order linear equation L[y] = g(t) are $\psi_{1}(t)=3 e^{t}+e^{t^{2}}, \psi_{2}(t)=7 e^{t}+e^{t^{2}}$ and $\psi_{3}(t)=5 e^{t}+e^{-t^{3}}+e^{t^{2}}.$ Find the solution of the initial-value problem $L[y]=g ; \quad y(0)=1, \quad y^{\prime}(0)=2.$

Write the general form of a second-order nonhomogeneous linear differential equation with constant coefficients.

Use an integral to find the specified area. Between y=cos x+7 and $y=\ln (x-3), 5 \leq x \leq 7$.

Harley-Davidson Inc. manufactures motorcycles. During the years following 2003 (the company's $100^{th}$ anniversary), the company's net revenue can be approximated by 4.6+0.4t billion dollars per year, where t is time in years since January 1, 2003. Assume this rate holds through January 1, 2014, and assume a continuous interest rate of 3.5% per year. What was the net revenue of the Harley-Davidson Company in 2003? What is the projected net revenue in 2013?

Determine whether each sequence converges or diverges. If it converges, find its limit. $\left\{1-\left(\frac{1}{2}\right)^{n}\right\}$

True or False. If f(x) is a related function of the sequence $\left\{s_{n}\right\}$ and there is a real number L for which $\lim _{x \rightarrow \infty} f(x)=L$ then $\left\{s_{n}\right\}$ converges.

Determine whether each sequence is bounded from above, bounded from below, both, or neither. $\left\{\frac{n^{2}}{n+1}\right\}$

Determine whether each sequence converges or diverges. If it converges, find its limit. $\left\{\cos \left(\frac{n}{e^{n}}\right)\right\}$

A third-order homogeneous linear equation and three linearly independent solutions are given. Find a particular solution satisfying the given initial conditions.

$$\left. \begin{array} { l } { y ^ { ( 3 ) } + 9 y ^ { \prime } = 0 ; y ( 0 ) = 3 , y ^ { \prime } ( 0 ) = - 1 , y ^ { \prime \prime } ( 0 ) = 2 ; y _ { 1 } = 1 } \\ { y _ { 2 } = \cos 3 x , y _ { 3 } = \sin 3 x } \end{array} \right.$$

Evaluate the limits or explain why they do not exist. $\lim _{x \rightarrow 0} x \sin \frac{1}{x}$

Using the definition of derivative , find equations for the tangent lines to the curves at the points indicated. $y=\sqrt{x+6}$ at the point (3, 3)

Find the general solution of each of the following equations. $\frac{d^{2} y}{d t^{2}}-3 \frac{d y}{d t}+2 y=t e^{3 t}+1$

Assume f satisfies the partial differential equation $\frac{\partial^{2} f}{\partial x^{2}}-2 \frac{\partial^{2} f}{\partial x \ \partial y} - 3 \frac{\partial^{2} f}{\partial y^{2}}=0$. Introduce the linear change of variables, $\mathrm{x}=A u + B v, \ y=C u + D v$, where A, B, C, D are constant, and let $g(u, \mathrm{v})=f(A u +B v, C u + D v)$. Compute nonzero integer values of A, B, C, D such thatg satisfies $\partial^{2} g /(\partial u \partial v)=0$. Solve this equation forg and thereby determinef. (Assume equality of the mixed partials.)

Find the area of the region R specified. It is helpful to make a sketch of the region. Under $y=e^{-x}$ and above y = 0 from x = -a to x = 0