082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

19.5 Second-order linear equations 559
thesameasitsgeneralsolution;weshalloftenwritey(x)forboth.Ifconditionsaregiven
theyareappliedtothegeneralsolutionoftheinhomogeneousequationtodetermineany
unknown constants. This yields the particular solution satisfyingthe given conditions.
Example19.15 Verify that y 1
(x) = x and y 2
(x) = 1 both satisfy d2 y
= 0. Write down the general
dx2 solution of thisequation and verifythatthis indeed satisfies the equation.
Solution Ify 1
(x) = x then dy 1
y 1
dx =1andd2 = 0, so thaty
dx 2 1
satisfies d2 y
dx = 0.Ify 2 2
(x) = 1,
then dy 2
y 2
dx =0andd2 = 0, so thaty
dx 2 2
satisfies d2 y
= 0. From Property 1, the general
dx2 solution of d2 y
dx =0is 2
y H
(x) =Ax +B(1)
=Ax+B
To verifythat thissatisfies the equation proceed as follows:
dy H
dx =A
d 2 y H
dx 2 = 0
and soy H
(x) satisfies d2 y
dx 2 = 0.
Example19.16 Given
d 2 y
+y=x (19.9)
dx2 (a) Show thaty H
= Acosx +Bsinx is a solution of the corresponding homogeneous
equation.
(b) Verifythaty P
=xisaparticular integral.
(c) Verifythaty H
+y P
does indeed satisfythe inhomogeneous equation.
Solution (a) Ify H
=Acosx +Bsinx, then
y ′ H =−Asinx+Bcosx
y′′ H =−Acosx−Bsinx
We see immediately thaty H
+y ′′ H = 0 so thaty H
is a solution of the homogeneous
equation.
(b) Ify P
= x theny ′ P = 1 andy′′ P
= 0. Substitution into the inhomogeneous equation
shows thaty P
satisfies Equation (19.9),thatisy P
=xisaparticular integral.
(c) Writing
we have
y=Acosx+Bsinx+x
y ′ =−Asinx+Bcosx+1
y ′′ =−Acosx−Bsinx