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Chapter 4: Q37E (page 186)

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

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

Short Answer

Expert verified

The particular solution isyp(t)=t2+3t-1.

Step by step solution

01

Consider the particular solution for the given differential equation.

The given differential equation is,

y'''-2y''-y'+2y=2t2+4t-9鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌�......(1)

Consider the particular solution is,

yp(t)=At2+Bt+C鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆�夆赌夆�夆赌夆�夆赌夆�夆赌夆�夆赌夆�夆赌�....(2)

Take the first, second, and third derivatives of the above equation,

yp'(t)=2At+Byp''(t)=2Ayp'''(t)=0

Substitute value of yp'(t),鈥夆赌�yp''(t)and yp'''(t)in the equation (1),

y'''-2y''-y'+2y=2t2+4t-90-2[2A]-[2At+B]+2[At2+Bt+C]=2t2+4t-92At2+(-2A+2B)t+(-4A-B+2C)=2t2+4t-9

Comparing all coefficients of the above equation;

2A=2鈬�A=1-2A+2B=4鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆�夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆�夆赌夆�夆赌夆�夆赌�......(3)-4A-B+2C=-9鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆�夆赌夆�夆赌夆�夆赌夆�夆赌夆�夆赌夆��......(4)

Substitute the value A in the equation (3),

-2(1)+2B=42B=6B=3

Substitute the value A and B in the equation (4),

-4(1)-3+2C=-92C=-2C=-1

02

Conclusion

Therefore, the particular solution of the equation (1),

yp(t)=At2+Bt+Cyp(t)=(1)t2+(3)t+(-1)yp(t)=t2+3t-1

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Most popular questions from this chapter

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.2y''-y'+6y=t2e-tsint-8tcos3t+10t

Solve the given initial value problem. y''-2y'+2y=0;y(蟺)=e蟺,y'(蟺)=0

Given that y1(t)=costis a solution to y''-y'+y=sintand y2(t)=e2t3is a solution to role="math" localid="1654926813168" y''-y'+y=e2t. Use the superposition principle to find solutions to the following differential equations:

(a)鈥夆赌夆�夆赌�y''-y'+y=5sint

(b)鈥夆赌夆�夆赌�y''-y'+y=sint-3e2t

(c)鈥夆赌夆�夆赌�y''-y'+y=4sint+18e2t

The auxiliary equations for the following differential equations have repeated complex roots. Adapt the "repeated root" procedure of Section 4.2 to find their general solutions:

(a)y''''+2y''+y=0

(b)y''''+4y'''+12y''+16y'+16y=0

Find the solution to the initial value problem.

y''(x)-y'(x)-2y(x)=cosx-sin2x;鈥夆赌夆�夆赌夆��y(0)=-720,鈥夆赌夆�夆赌�y'(0)=15
See all solutions

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