The Computable Differential Equation Lecture ... - Bruce E. Shapiro
The Computable Differential Equation Lecture ... - Bruce E. Shapiro The Computable Differential Equation Lecture ... - Bruce E. Shapiro
196 CHAPTER 10. APPENDIX ON ANALYTIC METHODS (MATH 280 IN A NUTSHELL) General Non-homogeneous Linear Equation with Constant Coefficients To solve ay ′′ + by ′ + cy = f(t) where a, b, c are constants for a general function f(t), the solution is ∫ ∫ ∫ y = Ae r1t + Be r 1t e r 2−r 1 sds + er 1t e r 2−r 1 s e −r2u f(u)duds a t where r 1 and r 2 are the roots of ar 2 + br + c = 0. An alternative method is to factor the equation into the form and make the substitution (D − r 1 )(D − r 2 )y = f(t) z = (D − r 2 )y This reduces the second order equation in y to a first order linear equation in z. Solve the equation (D − r 1 )z = f(t) for z, then solve the equation for y once z is known. Method of Reduction of Order (D − r 2 )y = z If one solution y 1 is known for the differential equation y ′′ + p(t)y ′ + q(t)y = 0 then a second solution is given by ∫ W (y1 , y 2 ))(t) y 2 (t) = y 1 (t) y 1 (t) 2 dt where the Wronskian is given by Abel’s formula ( ∫ W (y 1 , y 2 )(t) = Cexp − t s ) p(s)ds Method of Variation of Parameters To find a particular solution to y ′′ + p(t)y ′ + q(t)y = r(t) when a pair of linearly independent solutions to the homogeneous equation are already known, ∫ y p = −y 1 (t) t y ′′ + p(t)y ′ + q(t)y = 0 ∫ y 2 (s)r(s) W (y 1 , y 2 )(s) ds + y y 1 (s)r(s) 2(t) t W (y 1 , y 2 )(s) ds Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 10. APPENDIX ON ANALYTIC METHODS (MATH 280 IN A NUTSHELL) 197 Power Series Solution To solve y ′′ + p(t)y ′ + q(t)y = g(t) expand y, p, q and g in power series about ordinary (non-singular) points and determine the coefficients by applying linear independence to the powers of t. To solve a(t)y ′′ + b(t)y ′ + c(t)y = g(t) about a point t 0 where a(t) = 0 but the limits b(t)/a(t) and c(t)/a(t) exist as t → 0 (a regular singularity), solve the indicial equation r(r − 1) + rp 0 + q 0 = 0 for r where p 0 = lim t→0 b(t 0 )/a(t 0 ) and and q 0 = lim t→0 c(t 0 )/a(t 0 ). solution to the homogeneous equation is Then one y(t) = (t − t 0 ) r ∞ ∑ k=0 c k (t − t 0 ) k for some unknown coefficients c k .Determine the coefficients by linear independance of the powers of t. The second solution is found by reduction of order and the particular solution by variation of parameters. c○2007, B.E.Shapiro Last revised: May 23, 2007 Math 582B, Spring 2007 California State University Northridge
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CHAPTER 10. APPENDIX ON ANALYTIC METHODS<br />
(MATH 280 IN A NUTSHELL) 197<br />
Power Series Solution<br />
To solve<br />
y ′′ + p(t)y ′ + q(t)y = g(t)<br />
expand y, p, q and g in power series about ordinary (non-singular) points and determine<br />
the coefficients by applying linear independence to the powers of t.<br />
To solve<br />
a(t)y ′′ + b(t)y ′ + c(t)y = g(t)<br />
about a point t 0 where a(t) = 0 but the limits b(t)/a(t) and c(t)/a(t) exist as t → 0<br />
(a regular singularity), solve the indicial equation<br />
r(r − 1) + rp 0 + q 0 = 0<br />
for r where p 0 = lim t→0 b(t 0 )/a(t 0 ) and and q 0 = lim t→0 c(t 0 )/a(t 0 ).<br />
solution to the homogeneous equation is<br />
<strong>The</strong>n one<br />
y(t) = (t − t 0 ) r<br />
∞ ∑<br />
k=0<br />
c k (t − t 0 ) k<br />
for some unknown coefficients c k .Determine the coefficients by linear independance<br />
of the powers of t. <strong>The</strong> second solution is found by reduction of order and the<br />
particular solution by variation of parameters.<br />
c○2007, B.E.<strong>Shapiro</strong><br />
Last revised: May 23, 2007<br />
Math 582B, Spring 2007<br />
California State University Northridge
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