/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q5E In Problems \(1 - 8\( state the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Q5E (page 494)

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)\(.

\({\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^2} = \sqrt {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} \(

Short Answer

Expert verified

The equation is nonlinear and second order.

Step by step solution

01

Classification of linearity.

If \(F\(is linear in \(y,y',...,{y^n}\(, then the \({n^{th}}\(order ordinary differential equation is said to be linear. The form of the equation is given by,

\({a_n}(x)\frac{{{d^n}y}}{{d{x^n}}} + {a_{n - 1}}(x)\frac{{{d^{n - 1}}y}}{{d{x^{n - 1}}}} + L + {a_1}(x)\frac{{dy}}{{dx}} + {a_0}(x)y = g(x)\(

02

Determine whether it is linear or nonlinear.

As, by the classification of linearity, the given differential equation should be in the form,\({a_2}(x)\frac{{{d^2}y}}{{d{x^2}}} + {a_1}(x)\frac{{dy}}{{dx}} + {a_0}(x)y = g(x)\(.

Square the given equation to get,

\(\begin{aligned}{l}{\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^2} = 1 + {\left( {\frac{{dy}}{{dx}}} \right)^2}\\{\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^2} - {\left( {\frac{{dy}}{{dx}}} \right)^2} = 1\end{aligned}\(

But the term \(\frac{{dy}}{{dx}}\( and \(\frac{{{d^2}y}}{{d{x^2}}}\( has to be the power of \(2\(. So, the given is nonlinear.

If the given equation is linear, then the term \(\frac{{dy}}{{dx}}\( and \(\frac{{{d^2}y}}{{d{x^2}}}\( must have to be the power of \(1\(. Because of \(\frac{{{d^2}y}}{{d{x^2}}}\(, the equation is second order differential equation.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In Problems 21鈥�24 verify that the indicated family of functions is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution.

\(\frac{{dP}}{{dt}} = P(1 - P);\;P = \frac{{{c_1}{e^t}}}{{1 + {c_1}{e^t}}}\)

In Problems \(15 - 18\) verify that the indicated function \(y = \phi (x)\) is an explicit solution of the given first-order differential equation. Proceed as in Example \(6\), by considering \(\phi \) simply as a function and give its domain. Then by considering \(\phi \) as a solution of the differential equation, give at least one interval \(I\) of definition.

\((y - x)y' = y - x + 8;y = x + 4\sqrt {x + 2} \)

In Problems \(15\) and \(16\) interpret each statement as a differential equation.

On the graph of \(y = \phi (x)\) the slope of the tangent line at a point \(P(x,y)\) is the square of the distance from \(P(x,y)\) to the origin.

(a) Verify that the one-parameter family \({y^2} - 2y = {x^2} - x + c\) is an implicit solution of the differential equation \((2y - 2)y' = 2x - 1\).

(b) Find a member of the one-parameter family in part (a) that satisfies the initial condition \(y(0) = 1\).

(c) Use your result in part (b) to and an explicit function \(y = \phi (x)\) that satisfies \(y(0) = 1\). Give the domain of the function \(\phi \). Is \(y = \phi (x)\) a solution of the initial-value problem? If so, give its interval \(I\) of definition; if not, explain.

In Problems \(5\) and \(6\) compute \(y'\) and \(y''\) and then combine these derivatives with \(y\) as a linear second-order differential equation that is free of the symbols \({c_1}\) and \({c_2}\) and has the form \(F(y',y'',y''') = 0\). The symbols \({c_1}\) and \({c_2}\) represent constants.

\(y = {c_1}{e^x} + {c_2}x{e^x}\)

See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.