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Chapter 4: Q34E (page 186)

Determine the form of a particular solution for the differential equation. Do not solve

y''+5y'+6y=sint-cos2t

Short Answer

Expert verified

Thus, the answer is:yp=Asint+Bcost+Ccos2t+Dsin2t

Step by step solution

01

Firstly, write the auxiliary equation of the given differential equation

The differential equation is,

y''+5y'+6y=sint-cos2t鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�...1

Write the homogeneous differential equation of equation (1),

y''+5y'+6y=0

The auxiliary equation for the above equation,

m2+5m+6=0

02

Now find the complementary solution of the given equation is,

Solve the auxiliary equation,

m2+5m+6=0m2+3m+2m+6=0mm+3+2m+3=0m+2m+3=0

The roots of the auxiliary equation are,

m1=-2,鈥�鈥�鈥�鈥�鈥�鈥�m2=-3

The complementary solution of the given equation is,

yc=c1e-2t+c2e-3t

03

Now find the form of a particular solution

The particular solution to equation (1) will be the linear combination of termssint,鈥�鈥�cos2t and their independent derivatives.

Therefore, the particular solution of equation (1),

yp=Asint+Bcost+Ccos2t+Dsin2t

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Most popular questions from this chapter

Decide whether or not the method of undetermined coefficients can be applied to find a particular solution of the given equation. ty''-y'+2y=sin3t

Solve the given initial value problem. w''-4w'+2w=0;w(0)=0,w'(0)=1

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

Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equation. Do not solve the equation. y''+y'+ty=et+7

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).
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