20
1 Mathematical Physics
Procedure:
First step: Replace the RHS member of the given equation (I ) by zero and solve
the complimentary function of I to get y = u.
Second step: Differentiate successively the given equation (I ) and obtain, either
directly or by elimination, a differential equation of a higher order of type I.
Third step: Solving this new equation by the previous rule we get its complete
solution
y = u + v
where the part u is the complimentary function of (I ) already found in the first step,
and v is the sum of additional terms found
Fourth step: To find the values of the constants of integration in the particular
solution v, substitute
y = u + v
and its derivatives in the equation (I ). In the resulting identity equation equate the
coefficients of like terms, solve for constants of integration, substitute their values
back in
y = u + v
giving the complete solution of (I ).
Type III
d
n y
dx n = X
where X is a function of x alone, or constant
Integrate n times successively. Each integration will introduce one arbitrary
constant.
Type IV
d
2 y
dx 2 = Y
where Y is a function of y alone
Multiply the LHS member by the factor 2
dy
dx
dx and the RHS member by k equivalent factor 2dy
2
dy
dx
d
2 y
dx 2 dx = d
dy
dx
2
= 2Y dy
d
dy
dx
2
=
dy
dx
2
=
2Y dy + C 1