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Chapter 2: Q2E (page 46)

In problem 1-6,determine whether the differential equation is separabledydx=4y2-3y+1.

Short Answer

Expert verified

The differential equationdydx=4y2-3y+1 is separable.

Step by step solution

01

Concept Separable Differential Equation

A first-order ordinary differential equation dydx=f(x,y)is referred to as separable if the function in the right-hand side of the equation is expressed as a product of two functions g(x)g(x)that is a function of x alone and h(y)that is a function of yalone.

Mathematically, the equation dydx=f(x,y)is separable, when f(x)=g(x)h(y)..

02

Determining whether the equation is Separable or not

The given equation is

ddx=4y2-3y+1........(1)

The function in the right – hand side of equation (1) is

fx,y=4y2-3y+1=14y2-3y+1=gxhy

This function can be written as a product of two functions gxand hydefined as,

gx=1hy=4y2-3y+1

Therefore, the given differential equation is separable.

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Most popular questions from this chapter

In problem, solve the equation.(x+xy2)dx+ex2ydy=0

Question: Consider the initial value problem dydx+1+sin2xy=x,y(0)=2.

(a)Using definite integration, show that the integrating factor for the differential equation can be written as μ(x)=(∫0x1+sin2tdt) and that the solution to the initial value problem is y(x)=1μ(x)∫0xμ(s)sds+2μ(x)

(b)Obtain an approximation to the solution at x= 1 by using numerical integration (such as Simpson’s rule, Appendix C) in a nested loop to estimate values ofμ(x)and, thereby, the value of∫01μ(s)ds.

[Hint:First, use Simpson’s rule to approximateμ(x)at x= 0.1, 0.2, . . . , 1. Then use these values and apply Simpson’s rule again to approximate∫01μ(s)ds]

(c)Use Euler’s method (Section 1.4) to approximate the solution at x= 1, with step sizes h= 0.1 and 0.05. [A direct comparison of the merits of the two numerical schemes in parts (b) and (c) is very complicated, since it should take into account the number of functional evaluations in each algorithm as well as the inherent accuracies.]

Question: In Problems 1-30, solve the equation.

dydx+xy=0

Use the method discussed under “Homogeneous Equations” to solve problems 9-16.

(3x2-y2)dx+(xy-x3y-1)dy=0

x≡0In problems 33-40, Solve the equation given in Problem 2.

y-4x-12dx-dy=0
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