18
1 Mathematical Physics
Type IV equations reducible to linear form
Some equations that are not linear can be reduced to the linear form by a suitable
substitution, for example
( A)
dy
dx
+ Py = Qy
n
where P, Q are functions of x alone, or constants. Equation ( A) may be reduced to
the linear form (A), Type III by means of the substitution x = y
−n+1 .
Differential equations of the nth Order and of the nth degree
Consider special cases of linear differential equations.
Type I – The linear differential equation
(A)
d
n y
dx n + P 1
d
n−1 y
dx n−1 + P 2
d
n−2 y
dx n−2 + · · · + P n y = 0
in which coefficients P 1 , P 2 , . . . P n are constants.
Consider the differential equation of third order
(B)
d
3 y
dx 3 + P 1
d
2 y
dx 2 + P 2
dy
dx
+ P 3 y = 0
where P 1 , P 2 and P 3 are constants. The corresponding auxiliary equation is
r
3
+ P 1 r
2
+ P 2 r + P 3 = 0
Let the roots be r 1 , r 2 , r 3 .
If r 1 , r 2 , r 3 are real and distinct,
y = C 1 e
r1x
+ C 2 e
r2 x
+ C 3 e
r3 x
If r 1 , r 2 , r 3 are real and equal
y = C 1 e
−r 1 x
+ C 2 xe
−r 2 x
+ C 3 x
2 e
−r 3 x
In case a + bi and a − bi are each multiple roots of the auxiliary equation occurring s times, the solutions would be
C 1 e
ax cos bx, C 2 xe
ax cos bx, C 3 x
2 e
ax cos bx, . . . C s x
s−1 e
ax cos bx
C
1 e
ax sin bx, C
2 xe
ax sin bx, C
3 x
2 e
ax sin bx, . . . C
s x
s−1 e
ax sin bx
Summary for the rule for solving differential equations of the type
d
n y
dx n + P 1
d
n−1 y
dx n−1 + P 2
d
n−2 y
dx n−2 + · · · + P n y = 0
where P 1 , P 2 , . . . P n are constants.