Question 1

The complementary function of $y''+4y'+4y=0$ is

Accepted Answer

$(C_1+C_2x)e^{-2x}$

Answer

$e^{-4x}(C_1+C_2x)$

Answer

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

Answer

$C_1\cos2x+C_2\sin2x$

Question 2

A PI of $y''+y=e^x$ is

Accepted Answer

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

Answer

$xe^x$

Answer

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

Answer

$e^x$

Question 3

Using $D=\dfrac{d}{dx}$, a PI of $(D+4)y=e^{-4x}$ is

Accepted Answer

$xe^{-4x}$

Answer

$-xe^{-4x}$

Answer

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

Answer

$e^{-4x}$

Question 4

The CF of $(D^2-6D+13)^2y=0$ is

Accepted Answer

$e^{3x}[(C_1+C_2x)\cos2x+(C_3+C_4x)\sin2x]$

Answer

$(C_1+C_2x)e^{3x}+C_3\cos2x+C_4\sin2x$

Answer

$e^{3x}(C_1\cos2x+C_2\sin2x)$

Answer

$e^{-3x}[(C_1+C_2x)\cos2x+(C_3+C_4x)\sin2x]$

Question 5

For $x>0$, solve $x^2y''+xy'+4y=0$.

Accepted Answer

$C_1\cos(2\ln x)+C_2\sin(2\ln x)$

Answer

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

Answer

$x^2(C_1+C_2\ln x)$

Answer

$C_1x^2+C_2x^{-2}$

Question 6

For $x>-3$, solve $(x+3)^2y''-3(x+3)y'+4y=0$.

Accepted Answer

$(x+3)^2(C_1+C_2\ln(x+3))$

Answer

$C_1\cos(2\ln(x+3))+C_2\sin(2\ln(x+3))$

Answer

$(x+3)^{-2}(C_1+C_2\ln(x+3))$

Answer

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

Question 7

The Wronskian $W(\cos x,\sin x)$ equals

Accepted Answer

$1$

Answer

$\cos^2x-\sin^2x$

Answer

$0$

Answer

$-1$

Question 8

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 9

A spring-mass model satisfies $y''+16y=0$, $y(0)=3$, $y'(0)=0$. Its maximum displacement is

Accepted Answer

$3$

Answer

$4$

Answer

$16$

Answer

$12$

Question 10

A PI of $y''-9y=e^{3x}$ is

Accepted Answer

$\dfrac{xe^{3x}}{6}$

Answer

$\dfrac{e^{3x}}{6}$

Answer

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

Answer

$\dfrac{xe^{3x}}{3}$

Question 11

A PI of $(D+1)^3y=e^{-x}$ is

Accepted Answer

$\dfrac{x^3e^{-x}}{6}$

Answer

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

Answer

$xe^{-x}$

Answer

$\dfrac{e^{-x}}{3}$

Question 12

A PI of $y''+4y=8$ is

Accepted Answer

$2$

Answer

$0$

Answer

$4$

Answer

$8$

Question 13

A PI of $y''+9y=\sin3x+e^x$ is

Accepted Answer

$-\dfrac{x\cos3x}{6}+\dfrac{e^x}{10}$

Answer

$-\dfrac{x\sin3x}{6}+\dfrac{e^x}{10}$

Answer

$\dfrac{x\sin3x}{6}+\dfrac{e^x}{8}$

Answer

$\dfrac{x\cos3x}{6}+\dfrac{e^x}{10}$

Question 14

The fundamental period of $f(x)=\sin2x+\cos\left(\dfrac{3x}{2}\right)$ is

Accepted Answer

$4\pi$

Answer

$\pi$

Answer

$2\pi$

Answer

$\dfrac{4\pi}{3}$

Question 15

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

Accepted Answer

$0$

Answer

$\pi$

Answer

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

Answer

$a_n$

Question 16

For $f(x)=x^2$ on $(-\pi,\pi)$, $a_0$ equals

Accepted Answer

$\dfrac{2\pi^2}{3}$

Answer

$0$

Answer

$\pi^2$

Answer

$\dfrac{\pi^2}{3}$

Question 17

For $f(x)=x^2$ on $(-\pi,\pi)$, $a_2$ equals

Accepted Answer

$1$

Answer

$-1$

Answer

$0$

Answer

$4$

Question 18

Let $f(x)=4$ for $0<x<\pi$ and $f(x)=0$ for $-\pi<x<0$, extended periodically. The Fourier series value at $x=0$ is

Accepted Answer

$2$

Answer

$4$

Answer

undefined

Answer

$0$

Question 19

For the half-range sine series of $f(x)=x$ on $(0,\pi)$, $b_2$ equals

Accepted Answer

$-1$

Answer

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

Answer

$0$

Answer

$1$

Question 20

For the half-range cosine series of $f(x)=x^2$ on $(0,\pi)$, $a_1$ equals

Accepted Answer

$-4$

Answer

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

Answer

$0$

Answer

$4$

Question 21

In the complex Fourier series of $f(x)=\sin2x$, the nonzero coefficients are

Accepted Answer

$c_2=-\dfrac{i}{2},\;c_{-2}=\dfrac{i}{2}$

Answer

$c_{-2}=1$ only

Answer

$c_2=c_{-2}=\dfrac12$

Answer

$c_2=1$ only

Question 22

Applying Parseval's identity to $f(x)=x^2$ on $(-\pi,\pi)$ gives

Accepted Answer

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

Answer

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

Answer

$\sum_{n=1}^{\infty}\dfrac{(-1)^n}{n^2}=-\dfrac{\pi^2}{12}$

Answer

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

Question 23

For $f(x)=x$ on $(-1,1)$, expanded with period $2$, $b_3$ equals

Accepted Answer

$\dfrac{2}{3\pi}$

Answer

$-\dfrac{2}{3\pi}$

Answer

$0$

Answer

$\dfrac{2}{\pi}$

Question 24

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

Accepted Answer

only odd cosine terms

Answer

only even cosine terms

Answer

both all sine and all cosine terms

Answer

only odd sine terms

Question 25

For $f(x)=\operatorname{sgn}(\sin x)$, the coefficient of $\sin9x$ is

Accepted Answer

$\dfrac{4}{9\pi}$

Answer

$\dfrac{2}{9\pi}$

Answer

$-\dfrac{4}{9\pi}$

Answer

$0$

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