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Chapter 12: Q51E (page 526)

In Problems \(3\) and \(4\) Fill in the blank and then write this result as a linear second-order differential equation that is free of the symbols \({c_1}\) and \({c_2}\) and has the form \(F(y',y'') = 0\). The symbols \({c_1}\), \({c_2}\), and \(k\) represent constants.

\(\frac{{{d^2}}}{{d{x^2}}}({c_1}coskx + {c_2}sinkx) = \_\_\_\_\_\)

Short Answer

Expert verified

The answer is \(y'' + {k^2}y = 0\).

Step by step solution

01

Define second derivative of a function.

The derivative of the derivative of a function f is known as the second derivative, or second order derivative, in calculus.

So, the second derivative, or the rate of change of speed with respect to time, can be used to determine the variation in speed of the car (the second derivative of distance travelled with respect to the time).

02

Determine the second derivative of the function.

Let the first derivative of the given function be,

\(\frac{d}{{dx}}\left( {{c_1}coskx + {c_2}sinkx} \right) = - k{c_1}sinkx + k{c_2}coskx\)

Let the second derivative of the given function be,

\(\begin{array}{c}\frac{{{d^2}}}{{d{x^2}}}\left( {{c_1}coskx + {c_2}sinkx} \right) = - {k^2}{c_1}coskx - {k^2}{c_2}sinkx\\ = - {k^2}\left( {{c_1}coskx + {c_2}sinkx} \right)\end{array}\)

Substitute \({c_1}coskx + {c_2}sinkx = y\) in the above differential equation.

\(\begin{array}{c}\frac{{{d^2}y}}{{d{x^2}}} = - {k^2}y\\\frac{{{d^2}y}}{{d{x^2}}} + {k^2}y = 0\\y'' + {k^2}y = 0\end{array}\)

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Most popular questions from this chapter

Falling Body In Problem \(23\), suppose \(r = R + s\), where \(s\) is the distance from the surface of the Earth to the falling body. What does the differential equation obtained in Problem 23 become when \(s\) is very small in ( comparison to \(R\)? (Hint: Think binomial series for \({(R + s)^{ - 2}} = {R^{ - 2}}{(1 + s/R)^{ - 2}}\)).

In Problems 7鈥�12 match each of the given differential equations with one or more of these solutions:

(a) \(y = 0\), (b) \(y = 2\), (c) \(y = 2x\), (d) \(y = 2{x^2}\)

\(xy' = 2x\)

Let It Snow The 鈥渟nowplow problem鈥� is a classic and appears in many differential equation鈥檚 texts, but it was probably made famous by Ralph Palmer Agnew:

One day it started snowing at a heavy and steady rate. A snowplow started out at noon, going \(2\) miles the first hour and \(1\) mile the second hour. What time did it start snowing?

Find the textbook Differential Equations, Ralph Palmer Agnew, McGraw-Hill Book Co., and then discuss the construction and solution of the mathematical model.

In Problems \(1 - 8\) state the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with \((6)\).

\((1 - x)y'' - 4xy' + 5y = cos x\)

In Example \(7\) we saw that \(y = {\phi _1}(x) = \sqrt {25 - {x^2}} \) and \(y = {\phi _2}(x) = - \sqrt {25 - {x^2}} \) are solutions of \(dy/dx = - x/y\) on the interval \(( - 5,5)\). Explain why the piecewise-defined function \(y = \left\{ {\begin{aligned}{*{20}{c}}{\sqrt {25 - {x^2}} }&{ - 5 < x < 0}\\{ - \sqrt {25 - {x^2}} ,}&{0 \le x < 5}\end{aligned}} \right.\) is not a solution of the differential equation on the interval \(( - 5,5)\).
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