differential equation

486 DIFFERENTIAL EQUATIONS
3.
d 2 y
+ 9y = 26e2x
dx2 [y = A cos 3x + B sin 3x + 2e 2x ]
4. 9 d2 y
dt 2 − 6dy dt + y = 12e t 3
[y = (At + B)e 1 3 t + 2 3 t2 e 1 3 t ]
In problems 5 and 6 find the particular solutions
of the given differential equations.
5. 5 d2 y
dx 2 + 9dy dx −2y = 3ex ; when x = 0, y = 1 4
and dy
dx = 0. [
y = 5
) (e −2x − e 1 5 x + 1 ]
44
4 ex
6.
d 2 y
dt 2 − 6dy dt + 9y = 4e3t ; when t = 0, y = 2
and dy
dt = 0 [y = 2e3t (1 − 3t + t 2 )]
51.5 Worked problems on differential
equations of the form
a d2 y
dx 2 + bdy + cy = f (x) where f (x)
dx
is a sine or cosine function
Problem 7. Solve the differential equation
2 d2 y
dx 2 + 3dy − 5y = 6 sin 2x.
dx
Using the procedure of Section 51.2:
(i) 2 d2 y
dx 2 +3dy −5y = 6 sin 2x in D-operator form
dx
is (2D 2 + 3D − 5)y = 6 sin 2x
(ii) The auxiliary equation is 2m 2 + 3m − 5 = 0,
from which,
(m − 1)(2m + 5) = 0,
i.e. m = 1orm =−2
5
(iii) Since the roots are real and different the C.F.,
u = Ae x + Be − 2 5 x .
(iv) Let the P.I., v = A sin 2x + B cos 2x (see
Table 51.1(d)).
(v) Substituting v = A sin 2x + B cos 2x into
(2D 2 + 3D − 5)v = 6 sin 2x gives:
(2D 2 + 3D − 5)(A sin 2x + B cos 2x) = 6 sin 2x.
D(A sin 2x + B cos 2x)
= 2A cos 2x − 2B sin 2x
D 2 (A sin 2x + B cos 2x)
= D(2A cos 2x − 2B sin 2x)
=−4A sin 2x − 4B cos 2x
Hence (2D 2 + 3D − 5)(A sin 2x + B cos 2x)
=−8A sin 2x − 8B cos 2x + 6A cos 2x
− 6B sin 2x − 5A sin 2x − 5B cos 2x
= 6 sin 2x
Equating coefficient of sin 2x gives:
−13A − 6B = 6 (1)
Equating coefficients of cos 2x gives:
6A − 13B = 0 (2)
6 × (1)gives : − 78A − 36B = 36 (3)
13 × (2)gives : 78A − 169B = 0 (4)
(3) + (4)gives : − 205B = 36
from which,
B = −36
205
Substituting B = −36 into equation (1) or (2)
205
gives A = −78
205
Hence the P.I., v = −78 36
sin 2x − cos 2x.
205 205
(vi) The general solution, y = u + v, i.e.
y = Ae x + Be − 2 5 x
− 2 (39 sin 2x + 18 cos 2x)
205
d 2 y
Problem 8. Solve + 16y = 10 cos 4x
dx2 given y = 3 and dy = 4 when x = 0.
dx