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Chapter 4: Q7E (page 186)

A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation. y''=2y+2tan3x,鈥夆赌夆赌夆赌夆赌夆赌�yp(x)=tanx

Short Answer

Expert verified

The general solution of the given differential equation isy=c1e2x+c2e-2x+tanx.

Step by step solution

01

Firstly, write the auxiliary equation of the given differential equation

The differential equation is,

y''=2y+2tan3xy''-2y=2tan3x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�夆赌夆赌�鈥�(1)

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

y''-2y=0

The auxiliary equation for the above equation,

m2-2=0

02

Now find the complementary solution of the given equation is

Solve the auxiliary equation,

m2-2=0m=卤2

The roots of the auxiliary equation are,

m1=2,鈥夆赌夆赌�&鈥夆赌夆赌�m2=-2

The complementary solution of the given equation is,

yc=c1e2x+c2e-2x

03

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

The given particular solution,

yp(x)=tanx

Therefore, the general solution is,

y=yc(x)+yp(x)y=c1e2x+c2e-2x+tanx

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Most popular questions from this chapter

In Problems 34, use the method of undetermined coefficients to find a particular solution to the given higher-order equation. 2y'''+3y''+y'-4y=e-t

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)鈥夆赌夆赌夆��y''-y'+y=5sint

(b)鈥夆赌夆赌夆��y''-y'+y=sint-3e2t

(c)鈥夆赌夆赌夆��y''-y'+y=4sint+18e2t

Find a general solution y''+4y'+8y=0

Find a general solution.y''-8y'+7y=0

The auxiliary equation for the given differential equation has complex roots. Find a general solution.y''-4y'+7y=0
See all solutions

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