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Chapter 6: Q3E (page 332)

Find a general solution for the differential equation with x as the independent variable.6z'''+7z''-z'-2z=0

Short Answer

Expert verified

Thus, the general solution to the given differential equation is;

z=C1e-x+C2ex2+C3e-2x3

Step by step solution

01

Write the auxiliary equation.

The given differential equation is;

6z'''+7z''-z'-2z=0

The auxiliary equation is.6m3+7m2-m-2=0

Simplify the auxiliary equation.

6m3+7m2-m-2=06m3+7m2-m-2=0(m+1)(6m2+m-2)=0

One of the roots is . m1=-1

Find the other roots of the auxiliary equation by solving the quadratic equation.

m=-1卤1+4(12)12m=-1卤4912m=-1卤712m=-1+712,鈥�-1-712m2=12,鈥�m3=-23

02

Write the general solution.

The roots are real and distinct; therefore the general solution to the given differential equation is given as:

z=C1em1x+C2em2x+C3em3xz=C1e(-1)x+C2e(12)x+C3e(-23)xz=C1e-x+C2ex2+C3e-2x3

Thus, the general solution to the given differential equation is;

z=C1e-x+C2ex2+C3e-2x3

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Most popular questions from this chapter

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

y(4)+2y'''+10y''+18y'+9y=0

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

y'''+2y''-8y'=0

Use the annihilator method to show that ifa0鈮�0in equation (4) and fxhas the form (17) f(x)=bmxm+bm-1xm-1+鈰�+b1x+b0, thenyp(x)=Bmrxm+Bm-1xm-1+鈰�+B1x+B0is the form of a particular solution to equation (4).

As an alternative proof that the functions er1x,er2x,er3x,...,ernxare linearly independent on (鈭�,-鈭�) when r1,r2,...rn are distinct, assume C1er1x+C2er2x+C3er3x+...+Cnernxholds for all x in (鈭�,-鈭�) and proceed as follows:

(a) Because the ri鈥檚 are distinct we can (if necessary)relabel them so that r1>r2>r3>...>rn.Divide equation (33) by to obtain C1+C2er2xer2x+C3er3xer3x+...+Cnernxernx=0Now let x鈫掆垶 on the left-hand side to obtainC1 = 0.(b) Since C1 = 0, equation (33) becomes

C2er2x+C3er3x+...+Cnernx= 0for all x in(鈭�,-鈭�). Divide this equation byer2x

and let x鈫掆垶 to conclude that C2 = 0.

(c) Continuing in the manner of (b), argue that all thecoefficients, C1, C2, . . . ,Cn are zero and hence er1x,er2x,er3x,...,ernxare linearly independent on(鈭�,-鈭�).

(a) Derive the form y(x)=A1ex+A2e鈭�x+A3cosx+A4sinx for the general solution to the equation y(4)=y, from the observation that the fourth roots of unity are 1, -1, i, and -i.

(b) Derive the form

y(x)=A1ex+A2e鈭�x/2cos(3x2)+A3e鈭�x/2sin(3x2)

for the general solution to the equation y(3)=yfrom the observation that the cube roots of unity are 1,ei2蟺3 , and e鈭�i2蟺3.
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