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Chapter 1: Q10E (page 1)

Find a general solution for the differential equation with x as the independent variable:

y'''+3y''−4y'−6y=0

Short Answer

Expert verified

The general solution for the differential equation with x as the independent variable is y(x)=c1e−x+c2e(−1−7)x+c3e(−1+7)x

Step by step solution

01

Auxiliary equation:

Consider the equation .y'''+3y''−4y'−6y=0

The associated auxiliary equation is whichr3+3r2−4r−6=(r+1)(r2+2r−6)=0 has r=−1,r=−1−7 and r=−1+7as solutions

02

General solution:

The general solution to the given equation is given by

y(x)=c1e−x+c2e(−1−7)x+c3e(−1+7)x

Hence the final solution is y(x)=c1e−x+c2e(−1−7)x+c3e(−1+7)x

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Most popular questions from this chapter

In Problems 9–20, determine whether the equation is exact.

If it is, then solve it.

2x+y1+x2y2dx+x1+x2y2-2ydy=0

In Problems 23-28, determine whether Theorem 1 implies that the given initial value problem has a unique solution.

dydx=3x-y-13,y(2)=1

In problems 1-4Use 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 0.1 (h = 0.1).

dydx=y(2-y),y(0)=3

(a) For the initial value problem (12) of Example 9. Show that ϕ1(x)=0 andϕ2(x)=(x-2)3are solutions. Hence, this initial value problem has multiple solutions. (See also Project G in Chapter 2.)

(b) Does the initial value problemy'=3y23,y(0)=10-7have a unique solution in a neighbourhood ofx=0?

(a) Show that y2+x-3=0 is an implicit solution todydx=-12y on the interval (-∞,3).

(b) Show thatxy3-xy3sinx=1 is an implicit solution todydx=xcosx+sinx-1y3x-xsinx on the interval (0,π2).
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