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Chapter 4: Q4E (page 186)

A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation. y''+y'=1,鈥夆赌夆赌夆赌夆赌夆赌�yp(t)=t

Short Answer

Expert verified

y=c1+c2e-t+t

Step by step solution

01

Write the auxiliary equation of the given differential equation.

The differential equation is,

y''+y'=1鈥夆赌夆赌夆赌夆赌夆赌夆�夆赌夆赌夆赌夆赌夆赌夆�夆赌夆赌夆赌夆赌夆赌夆�夆�夆��鈥�(1)

Write the homogeneous differential equation of the equation (1),

y''+y'=0

The auxiliary equation for the above equation,

m2+m=0

02

Now find the complementary solution of the given equation is 

Solve the auxiliary equation,

m2+m=0m(m+1)=0

The roots of the auxiliary equation are,

m1=0,鈥夆赌夆赌夆赌夆赌夆赌�m2=-1

The complementary solution of the given equation is,

yc=c1+c2e-t

03

Use the given particular solution to find a general solution for the equation

The given particular solution,

yp(t)=t

Therefore, the general solution is,

y=yc(t)+yp(t)y=c1+c2e-t+t

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Most popular questions from this chapter

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

Decide whether or not the method of undetermined coefficients can be applied to find a particular solution of the given equation. x''+5x'-3x=3t

Find a particular solution to the differential equation.

y''+4y=8sin(2t)

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Solve the given initial value problem y''+2y'+17y=0;y(0)=1,y'(0)=-1.

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