Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 1: Q6 E (page 14)

In Problems 3-8, determine whether the given function is a solution to the given differential equation.

x=cos2t,dxdt+tx=sin2t

Short Answer

Expert verified

The given function is not a solution to the given differential equation.

Step by step solution

01

Differentiating the given equation w.r.t. (with respect to) t.

Firstly, we will differentiate x=cos2twith respect to t,

dxdt=-2sin2t

02

Simplification.

Putting the values from step 1 in the L.H.S. (Left-hand side) of the given differential equation,

dxdt+tx=-2sin2t+tcos2t

which is not the same as the R.HS. (Right-hand side) of the given differential equation.

Hence,x=cos2tis not a solution to the differential equation dxdt+tx=sin2t.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

Let ϕ(x)denote the solution to the initial value problem

dydx=x-y,y(0)=1

⦁ Show that ϕ(x)=1-ϕ'(x)=1-x+ϕ(x)

⦁ Argue that the graph of ϕ is decreasing for x near zero and that as x increases from zero, ϕ(x)decreases until it crosses the line y = x, where its derivative is zero.

⦁ Let x* be the abscissa of the point where the solution curve y=ϕ(x) crosses the line y=x.Consider the sign of ϕ(x*) and argue that ϕ has a relative minimum at x*.

⦁ What can you say about the graph of y=ϕ(x) for x > x*?

⦁ Verify that y = x – 1 is a solution to dydx=x-y and explain why the graph of y=ϕ(x) always stays above the line y=x-1.

⦁ Sketch the direction field for dydx=x-y by using the method of isoclines or a computer software package.

⦁ Sketch the solution y=ϕ(x) using the direction field in part (f).

Question:(a) Use the general solution given in Example 5 to solve the IVP. 4x"+e-0.1tx=0,x(0)=1,x'(0)=-12.Also use J'0(x)=-J1(x) and Y'0(x)=-Y1(x)=-Y1(x)along withTable 6.4.1 or a CAS to evaluate coefficients.

(b) Use a CAS to graph the solution obtained in part (a) for.

In Problems 3–8, determine whether the given function is a solution to the given differential equation.

θ=2e3t-e2t,d2θdt2-θdθdt+3θ=-2e2t

Combat Model.A simplified mathematical model for conventional versus guerrilla combat is given by the systemx'1=-(0.1)x1x2;  x1(0)=10;  x'2=-x1;x2(0)=15 wherex1 andx2 are the strengths of guerrilla and conventional troops, respectively, and 0.1 and 1 are the combat effectiveness coefficients.Who will win the conflict: the conventional troops or the guerrillas? [Hint:Use the vectorized Runge–Kutta algorithm for systems with h=0.1to approximate the solutions.]

In problems 1-4 Use Euler’s method to approximate the solution to the given initial value problem at the points x=0.1,0.2,0.3,0.4, and 0.5, using steps of size 0.1h=0.1.

dydx=xy  ,  y(0)=-1
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

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.

Sign-up for free