ABSTRACT
Procedure to solve differential equations of the form a d 2 y dx 2 + b dy dx + cy = f ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4843.tif"/>
Rewrite the given differential equation as ( aD 2 + bD + c ) y = f ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4844.tif"/>
Substitute m for D, and solve the auxiliary equation am 2 + bm + c = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4845.tif"/> for m
Obtain the complementary function, u, which is achieved using the same procedure as on page 391–92
To determine the particular integral, v, firstly assume a particular integral which is suggested by f(x), but which contains undetermined coefficients. Table 141.1 gives some suggested substitutions for different functions f(x). Form of particular integral for different functions
Type
Straightforward cases Try as particular integral:
‘Snag’ cases Try as particular integral:
(a) f ( x ) = a constant https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4846.tif"/>
v = k https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4847.tif"/>
v = kx https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4848.tif"/> (used when C.F. contains a constant)
(b) f ( x ) = polynomial https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4849.tif"/> (i.e. f ( x ) = L + Mx + Nx 2 + .. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4850.tif"/> where any of the coefficients may be zero)
v = a + bx + cx 2 + .. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4851.tif"/>
(c) f ( x ) = an exponential function https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4852.tif"/> (i.e. f ( x ) = Ae ax https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4853.tif"/> )
v = ke ax https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4854.tif"/>
v = kxe ax https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4855.tif"/> (used when eax appears in the C.F.)
v = kx 2 e ax https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4856.tif"/> (used when eax and xeax both appear in the C.F.)
(d) f(x) a sine or cosine function (i.e. f(x) a sin px + b cos px where a or b may be zero)
v = A sin px + B cos px https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4857.tif"/>
v = x ( A sin px + B cos px ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4858.tif"/> (used when sin px and/or cos px appears in the C.F.)
(e) f ( x ) = a sum https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4859.tif"/> e.g.
f ( x ) = 4 x 2 − 3 sin 2 x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4860.tif"/>
f ( x ) = 2 − x + e 3 x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4861.tif"/>
v = ax 2 + bx + c + d sin 2x + e cos 2x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4862.tif"/>
v = ax + b + ce 3 x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4863.tif"/>
(f) f(x) = a product e.g. f ( x ) = 2 e x cos 2 x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4864.tif"/>
v = e x ( A sin 2x + B cos 2x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4865.tif"/>
Substitute the suggested P.I. into the differential equation ( aD 2 + bD + c ) v = f ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4866.tif"/> and equate relevant coefficients to find the constants introduced.
The general solution is given by y = C .F . + P .I . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4867.tif"/> i.e. y = u + v https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq4868.tif"/>
Given boundary conditions, arbitrary constants in the C.F. may be determined and the particular solution of the differential equation obtained.
