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Chapter 4: Q8E (page 164)

Question: find a general solution to the given differential equation.

z''+z'-z=0

Short Answer

Expert verified

Answer

The general solution of the given equation isz=c1e-1+52t+c2e-1-52t.

Step by step solution

01

Write the auxiliary equation of the given differential equation.Step 1: Write the auxiliary equation of the given differential equation.

The given differential equation isz''+z'-z=0.

The auxiliary equation for the above equationm2+m-1=0.

02

Now find the roots of the auxiliary equation.

Solve the auxiliary equation,

m2+m-1=0m=-1卤1-41-12m=-1卤52

The roots of the auxiliary equation are

m1=-1+52,鈥�&鈥�m2=-1-52.

03

Write the general solution.

If an auxiliary equation has distinct real roots as&, then the general solution is given as;

y=c1em1t+c2em2t

Thus, the general solution of the given equation isz=c1e-1+52t+c2e-1-52t.

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Most popular questions from this chapter

Find a particular solution to the given higher-order equation.

y'''-2y''-y'+2y=2t2+4t-9

Determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.) y''+9y=4t3sin3t

Determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.) y''+2y'+2y=8t3e-tsint

Prove the sum of angles formula for the sine function by following these steps. Fix x.

(a)Let f(t)=sin(x+t). Show that f''(t)+f(t)=0, the standard sum of angles formula forsin(x+t) .f(0)=sinx , and f'(0)=cosx.

(b)Use the auxiliary equation technique to solve the initial value problem y''+y=0,y(0)=sinx, andy'(0)=cosx

(c)By uniqueness, the solution in part(b) is the same as following these steps. Fix localid="1662707913644" x.localid="1662707910032" f(t) from part (a). Write this equality; this should be the standard sum of angles formula for sin(x+t).

The auxiliary equation for the given differential equation has complex roots. Find a general solution.4y''+4y'+6y=0
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