(a) Verify that the given functions are linearly independent and form a basis of solutions of the given ODE. (b) Solve the IVP. Graph or sketch the solution.

$$4x^2y''-3y=0, y(1)=-3, y(1)=0, x^{3/2},x^{-1/2}$$

Find and graph or sketch the solution of the IVP. Show the details. y''+3y'+2y=10(sin t+δ(t-1), y(0)=1, y'(0)=-1

(a) Verify that the given functions are linearly independent and form a basis of solutions of the given ODE. (b) Solve the IVP. Graph or sketch the solution. x²y''-xy'+y=0, y(1)=4.3, y'(1)=0.5, x, x ln x

(a) Verify that the given functions are linearly independent and form a basis of solutions of the given ODE. (b) Solve the IVP. Graph or sketch the solution.

Reduce to first order and solve, showing each step in detail. yy''=3y'²

Solve the IVP $y^{\prime \prime}+9 y=20 e^{-t} : \quad y(0)=0, \quad y^{\prime}(0)=1$ using Laplace transforms.

A long coaxial cable consists of two thin-walled concentric conducting cylinders with radii $a$ and $b$. The inner cylinder $A$ carries a steady current $i$, the outer cylinder $B$ providing the return path. $(a)$ Calculate the energy stored in the magnetic field between the cylinders for length $l$ of the cable. $(b)$ What is the stored energy per unit length of the cable if $a=1.2 \mathrm{~mm}$, $b=3.5 \mathrm{~mm}$, and $i=2.7 \mathrm{~A}$ ?

Reduce to first order and solve, showing each step in detail. 2xy''=3y'

y''-y'=0

Reduce to first order and solve, showing each step in detail. y''=1+y'²

Reduce to first order and solve, showing each step in detail. xy''+2y'+xy=0, y_1=(cos x)/x

Find a real general solution. Show the details of your work. 5x²y''+23xy'+16.2y=0

An experiment was conducted to compare the effectiveness of three training programs, A, B, and C, in training assemblers of a piece of electronic equipment. Fifteen employees were randomly assigned, five each, to the three programs. After completion of the program each person was required to assemble four pieces of the equipment, and the average length of time required to complete the assembly was recorded. Several of the employees resigned during the course of the program; the remainder were evaluated, producing the data shown in the accompanying table. Use the Excel printout to answer the questions.

Training Program A: 59 64 57 62 Training Program B: 52 58 54 Training Program C: 58 65 71 63 64

a. Do the data provide sufficient evidence to indicate a difference in mean assembly times for people trained by the three programs? Give the p-value for the test and interpret its value. b. Find a 99% confidence interval for the difference in mean assembly times between person trained by programs A and B. c. Find a 99% confidence interval for the mean assembly times for persons trained by program A. d. Do you think that the data will satisfy (approximately) the assumptions that they have been selected from normal populations? Why?

Anova: Single Factor Groups, Count, Sum, Average, Variance A, 4, 242, 60.5, 9.667 B, 3, 164, 54.667, 9.333 C, 5, 321, 64.2, 21.7 ANOVA Source of Variation, SS, df, MS, F, P-value, F crit Between Groups, 170.45, 2, 85.225, 5.704, 0.0251, 4.256 Within Groups, 134.467, 9, 14.941 Total, 304.917, 11

Show that F(y, y', y'')=0 can be reduced to a first-order ODE with y as the independent variable and y''=(dz/dy)z, where z=y'; derive this by the chain rule. Give two examples.