First Order Linear Differential Equations16



3.4 Second-Order Differential Equations

The second order differential equation with constant coefficient

Consider a second order differential equation of the form

[pic] = m2y, where m2 is a positive constant (3.4-8)

Assume that the solution is y = Aekx, where A and k are two unknown constants then

[pic]= Akekx, [pic] = Ak2ekx

The constant k can be obtained by substituting the solution and its second derivative into Eq. (3.4-8)

Ak2ekx = Am2ekx

Therefore k = m or k = ( m, and the most general solution of Eq. (3.4-8) is

y = A1emx + A2e-mx (3.4-9)

where A1 and A2 are the two constants of integration. Eq. (3.4-9) can also be expressed in terms of the hyperbolic functions sinh(mx) and cosh(mx)

sinh(mx) = [pic], cosh(mx) = [pic]

or

emx = sinh(mx) + cosh(mx), e-mx = ( sinh(mx) + cosh(mx)

y = (A1 - A2)sinh(mx) + (A1 + A2)cosh(mx)

y = B1sinh(mx) + B2cosh(mx) (3.4-10)

Eq. (3.4-9) is usually used when the range of x is infinite and Eq. (3.4-10) is usually used when the range of x is finite.

When the coefficient of y in Eq. (3.4-8) is negative, that is

[pic] = ( m2y = i2m2y, where i2 = (1 (3.4-11)

Then the solution to Eq. (3.4-11) is

y = A1eimx + A2e-imx (3.4-12)

Eq. (3.4-11) can be expressed in terms of the real functions by using the Euler’s identity

eimx = cos(mx) + isin(mx) , and e-imx = cos(mx) - isin(mx)

y = A1(cos(mx) + isin(mx)) + A2(cos(mx) - isin(mx))

y = (A1 + A2)cos(mx) + i(A1 - A2)sin(mx)

y = C1cos(mx) + C2sin(mx) (3.4-13)

The second order nonhomogeneous differential equation with constant coefficient

Eq. (3.4-8) can be written as

[pic] ( m2y = 0 (3.4-14)

where all the terms that contains the dependent variable y are moved to the left hand side of the equation. The right hand side is equal to zero. Eq. (3.4-14) is a homogeneous equation. If the RHS of the differential equation is not equal to zero, it is a nonhomogeneous equation.

[pic] ( m2y = f(x) (3.4-15)

Eq. (3.4-15) is a nonhomogeneous with forcing function f(x). Let consider the simplest case when f(x) is a constant K.

[pic] ( m2y = K (3.4-16)

The general solution, y, of Eq. (3.4-16) is simply the sum of the general solution, yc, of Eq. (3.4-14) and a particular solution, yp, of Eq. (3.4-16).

y = yc + yp

where yc = A1emx + A2e-mx

yc is called the complimentary solution of Eq. (3.4-15). A particular solution of Eq. (3.4-16) can be obtained by assuming yp = C = constant since the RHS of Eq. (3.4-16) is a constant. Substituting yp into Eq. (3.4-16)

( m2C = K or C = ( K/m2

Hence the general solution to Eq. (3.4-16) is

y = yc + yp = A1emx + A2e-mx ( K/m2 (3.4-17)

Eq. (3.4-17) contains two arbitrary constants A1 and A2 to be determined from the boundary conditions.

Example: Find the general solution to the differential equation

[pic] ( 4y = 3; y(0) = 0, y (1) = 2

Solution:

The complimentary solution of [pic] ( 4y = 3 is the general solution to the homogeneous equation [pic] ( 4y = 0

yc = A1e2x + A2e-2x

The particular solution of [pic] ( 4y = 3 is yp = C

( 4C = 3 ( C = ( 0.75

The general solution is then y = A1e2x + A2e-2x ( 0.75

at x = 0, y = 0 = A1 + A2 ( 0.75 ( A1 + A2 = 0.75

at x = 1, y = 2 = 7.3891 A1 + 0.1353A2 ( 0.75 ( 7.3891A1 + 0.1353A2 = 2.75

Solve for A1 and A2 to obtain A1 = 0.3651, and A2 = 0.3849. The final solution is

y = 0.3651e2x + 0.3849e-2x ( 0.75

>> y=dsolve('D2y-4*y=3','y(0)=0','y(1)=2','x')

y =

-3/4-1/4*(-11+3*cosh(2))/sinh(2)*sinh(2*x)+3/4*cosh(2*x)

>> y=simple(y)

y =

1/4*(-3*sinh(2)+11*sinh(2*x)-3*sinh(2*x-2))/sinh(2)

>> x=0:.1:1;

>> [x' (eval(y))']

|ans = | 0.5000 0.3841 |

|0 0 |0.6000 0.5782 |

|0.1000 0.0111 |0.7000 0.8256 |

|0.2000 0.0527 |0.8000 1.1362 |

|0.3000 0.1265 |0.9000 1.5225 |

|0.4000 0.2355 |1.0000 2.0000 |

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