Question 1

The complementary function of $y''+y'-2y=0$ is

Accepted Answer

$C_1e^x+C_2e^{-2x}$

Answer

$C_1e^{-x}+C_2e^{2x}$

Answer

$C_1\cos2x+C_2\sin2x$

Answer

$(C_1+C_2x)e^{-x}$

Question 2

A particular integral of $y''+16y=\cos4x$ is

Accepted Answer

$\dfrac{x\sin4x}{8}$

Answer

$\dfrac{x\cos4x}{8}$

Answer

$-\dfrac{\cos4x}{15}$

Answer

$-\dfrac{x\sin4x}{8}$

Question 3

Using $D=\dfrac{d}{dx}$, a PI of $(D-5)y=e^{5x}$ is

Accepted Answer

$xe^{5x}$

Answer

$\dfrac{e^{5x}}5$

Answer

$e^{5x}$

Answer

$5xe^{5x}$

Question 4

The CF of $(D^2+6D+10)^2y=0$ is

Accepted Answer

$e^{-3x}[(C_1+C_2x)\cos x+(C_3+C_4x)\sin x]$

Answer

$e^{3x}[(C_1+C_2x)\cos x+(C_3+C_4x)\sin x]$

Answer

$e^{-3x}(C_1\cos x+C_2\sin x)$

Answer

$C_1e^{-3x}+C_2xe^{-3x}+C_3\cos x+C_4\sin x$

Question 5

For $x>0$, solve $x^2y''-xy'+y=0$.

Accepted Answer

$x(C_1+C_2\ln x)$

Answer

$C_1x+C_2x^{-1}$

Answer

$(C_1+C_2x)e^x$

Answer

$C_1\cos(\ln x)+C_2\sin(\ln x)$

Question 6

For $x>-4$, solve $(x+4)^2y''+(x+4)y'+9y=0$.

Accepted Answer

$C_1\cos(3\ln(x+4))+C_2\sin(3\ln(x+4))$

Answer

$\dfrac{C_1+C_2\ln(x+4)}{x+4}$

Answer

$(x+4)^3(C_1+C_2\ln(x+4))$

Answer

$C_1(x+4)^3+C_2(x+4)^{-3}$

Question 7

The Wronskian $W(e^x,xe^x)$ equals

Accepted Answer

$e^{2x}$

Answer

$e^x$

Answer

$0$

Answer

$-e^{2x}$

Question 8

For $(D-2)x+y=0,\;x+(D-2)y=0$, the independent modes are

Accepted Answer

$e^t$ and $e^{3t}$

Answer

$e^{2t}$ and $te^{2t}$

Answer

$e^{-t}$ and $e^{-3t}$

Answer

$1$ and $e^{2t}$

Question 9

A spring-mass model satisfies $y''+25y=0$, $y(0)=0$, $y'(0)=10$. Its maximum displacement is

Accepted Answer

$2$

Answer

$5$

Answer

$10$

Answer

$25$

Question 10

A PI of $y''-16y=e^{-4x}$ is

Accepted Answer

$-\dfrac{xe^{-4x}}{8}$

Answer

$\dfrac{xe^{-4x}}{8}$

Answer

$\dfrac{e^{-4x}}{8}$

Answer

$\dfrac{x^2e^{-4x}}{2}$

Question 11

A PI of $(D-1)^2y=e^x$ is

Accepted Answer

$\dfrac{x^2e^x}{2}$

Answer

$xe^x$

Answer

$\dfrac{e^x}{2}$

Answer

$\dfrac{xe^x}{2}$

Question 12

A PI of $y''+9y=18$ is

Accepted Answer

$2$

Answer

$18$

Answer

$3$

Answer

$9$

Question 13

A PI of $y''+y=\sin x+e^{2x}$ is

Accepted Answer

$-\dfrac{x\cos x}{2}+\dfrac{e^{2x}}{5}$

Answer

$\dfrac{x\sin x}{2}+\dfrac{e^{2x}}{3}$

Answer

$-\dfrac{x\sin x}{2}+\dfrac{e^{2x}}{5}$

Answer

$\dfrac{x\cos x}{2}+\dfrac{e^{2x}}{5}$

Question 14

The fundamental period of $f(x)=\cos5x+\sin10x$ is

Accepted Answer

$\dfrac{2\pi}{5}$

Answer

$\pi$

Answer

$2\pi$

Answer

$\dfrac{\pi}{10}$

Question 15

If $f$ is odd on $(-\pi,\pi)$, then $\int_{-\pi}^{\pi} f(x)\cos nx\,dx$ equals

Accepted Answer

$0$

Answer

$2\int_0^\pi f(x)\cos nx\,dx$

Answer

$\pi$

Answer

$b_n$

Question 16

For $f(x)=x$ on $(-\pi,\pi)$, $b_4$ equals

Accepted Answer

$-\dfrac12$

Answer

$\dfrac{4}{\pi}$

Answer

$\dfrac12$

Answer

$0$

Question 17

For $f(x)=|x|$ on $(-\pi,\pi)$, $a_1$ equals

Accepted Answer

$-\dfrac{4}{\pi}$

Answer

$0$

Answer

$\dfrac{4}{\pi}$

Answer

$-\dfrac{2}{\pi}$

Question 18

A periodic function has left limit $3$ and right limit $-1$ at $x=a$. Its Fourier series value at $a$ is

Accepted Answer

$1$

Answer

$2$

Answer

$-1$

Answer

$3$

Question 19

For the half-range sine series of $f(x)=1$ on $(0,\pi)$, $b_5$ equals

Accepted Answer

$\dfrac{4}{5\pi}$

Answer

$0$

Answer

$-\dfrac{4}{5\pi}$

Answer

$\dfrac{2}{5\pi}$

Question 20

For the half-range cosine series of $f(x)=x$ on $(0,\pi)$, $a_1$ equals

Accepted Answer

$-\dfrac{4}{\pi}$

Answer

$-4$

Answer

$0$

Answer

$\dfrac{4}{\pi}$

Question 21

In the complex Fourier series of $f(x)=e^{-i5x}$, the coefficients are

Accepted Answer

$c_{-5}=1$, all others $0$

Answer

$c_5=1$, all others $0$

Answer

$c_0=1$, all others $0$

Answer

$c_5=c_{-5}=\dfrac12$

Question 22

Using the Fourier series of $\operatorname{sgn}(\sin x)$, Parseval gives

Accepted Answer

$\sum_{k=0}^{\infty}\dfrac1{(2k+1)^2}=\dfrac{\pi^2}{8}$

Answer

$\sum_{n=1}^{\infty}\dfrac1{n^4}=\dfrac{\pi^4}{90}$

Answer

$\sum_{k=1}^{\infty}\dfrac1{(2k)^2}=\dfrac{\pi^2}{8}$

Answer

$\sum_{n=1}^{\infty}\dfrac1{n^2}=\dfrac{\pi^2}{6}$

Question 23

For $f(x)=x$ on $(-4,4)$, expanded with period $8$, $b_4$ equals

Accepted Answer

$-\dfrac{2}{\pi}$

Answer

$\dfrac{2}{\pi}$

Answer

$0$

Answer

$\dfrac{4}{\pi}$

Question 24

If $f$ is even and satisfies $f(x+\pi)=-f(x)$, then the Fourier series contains

Accepted Answer

only odd cosine terms

Answer

only even cosine terms

Answer

only odd sine terms

Answer

all sine and cosine harmonics

Question 25

For $f(x)=\operatorname{sgn}(\cos x)$, the coefficient of $\cos5x$ is

Accepted Answer

$\dfrac{4}{5\pi}$

Answer

$\dfrac{2}{5\pi}$

Answer

$-\dfrac{4}{5\pi}$

Answer

$0$