103

CHAPTER 2

5.

Linearity and Nonlinearity

y′ + y =

SECTION 2.2

Solving the First-Order Linear Differential Equation

103

1 1 + et

We multiply each side of the equation by the integrating factor μ ( t ) = et , giving et

et ( y ′ + y ) =

d

, or,

1 + et

(

( ye ) = t

et

.

1 + et

dt

)

Integrating, we get yet = ln 1 + et + c .

Hence, 6.

(

)

y ( t ) = ce −t + e −t ln 1 + et .

y ′ + 2ty = t In this problem we see that y p ( t ) =

1 is a solution of the nonhomogeneous equation (there are 2

other single solutions, but this is the easiest to find). Hence, to find the general solution we solve the corresponding homogeneous equation, y′ + 2ty = 0 , by separation of variables, getting dy = −2tdt , y which has the general solution y = ce −t , where c is any constant. 2

Adding the solutions of the homogeneous equation to the particular solution y = p

1

we get the

2

general solution of the nonhomogeneous equation: y ( t ) = ce −t + 2

7.

1 . 2

y ′ + 3t 2 y = t 2 In this problem we see that y p ( t ) =

1 is a solution of the nonhomogeneous equation (there are 3

other single solutions, but this is the easiest to find). Hence, to find the general solution, we solve the corresponding homogeneous equation, y′ + 3t 2 y = 0 , by separation of variables, getting dy = −3t 2 dt , y which has the general solution y ( t ) = ce−t , where c is any constant. Adding the solutions of the 3

1 homogeneous equation to the particular solution y = , we get the general solution of the

Solutions manual for differential equations and linear algebra 2nd edition by farlow ibsn 9780134689

Solutions Manual for Differential Equations and Linear Algebra 2nd Edition by Farlow IBSN 9780134689548 Full clear download( no error format...

Solutions manual for differential equations and linear algebra 2nd edition by farlow ibsn 9780134689

Solutions Manual for Differential Equations and Linear Algebra 2nd Edition by Farlow IBSN 9780134689548 Full clear download( no error format...