Solutions of Linear Differential Equations

1 Appendix A Solutions of Linear Differential Equations Linear Differential Equations with Constant Coefficients Linear diflFerential Equations with constant coefficients are usually writ-ten as 2/("> + ai2/("-i) + .. + a _i2/(i) + anV = g, ( ) where a^, fc = 1,.., n, are numbers, y^^^ = ^ , and g = g{t) is a known function of t. We shall denote hy D = ^ the derivative operator^ so that the Differential equation now becomes p{D)y = (D^ + aiD^-i + .. + a^_iD + an)y = g. ( ) If g(t) = 0, the equation is said to be homogeneous. If g{t) ^ 0, then the homogeneous or reduced equation is obtained from ( ) by replacing g byO.}}}

2 If y and y* are two different Solutions of ( ), then it is easy to show that y y* solves the reduced equation of ( ). Hence, if y is any solution to ( ), it can be written as y = y*+y\ ( ) where y* is any other particular solution to ( ) and y^ is a suitable solution to the homogeneous equation. Therefore, solving ( ) involves (a) finding all the Solutions to the homogeneous equation, caUed the gen-eral solution, and (b) finding a particular solution to the given equation. 364 A. Solutions of Linear Differential Equations The rest of these notes indicate how to solve these two problems.

3 Given ( ) the auxiliary equation is p{m) = mP + aim^'^ + .. + an-im + an = 0, ( ) In other words, p{m) is obtained from p{D) by replacing D by m. The auxiliary equation is an ordinary polynomial of nth degree and has n real or complex roots, counting multiple roots according to their multiplicity. We will see that, given these roots, we can write the general solution forms of homogeneous Unear Differential Equations . Homogeneous Equations of Order One Here the equation is (D - a)y = y'-ay = 0, which has y = Ce^^ as its general solution form. Homogeneous Equations of Order Two Here the Differential equation can be factored (using the quadratic for-mula) as (D-mi)(Z)-m2)2/-0, where m\ and m^ can be real or complex.}}}

4 Examples are given in Table and the solution forms are given in Table Differential Equation 1. y"-Ay' + Ay = Q 2. y" - %' + 3y = 0 3. y" - 4y' + 5y = 0 y{t) y{t) y{t) General Solution Form = e2*(Ci + Cit) = e2*(Di sinh t) + D2 sinh t) = e^\Disva. t + Di cos t) Table : Examples of Homogeneous Equations of Order Two Homogeneous Equations of Order n 365 Root ^1 7^ ^2? real mi = a + 6, 1712 = a b mi = 7722 = ^^ ^1 7^ ^2 J complex mi = a + bi^ m2 = a bi General Solution Form y{t) = Cie^i* + C2e^2t = e^*(Cie^^ + C2e-^^) or y(t) = e"*(Ci sinh 6t + C2 cosh bt) y{t) = iCi+C2t)e^' y(t) = Cie^i* + C2e^2t - e"^(Cie^^* + C2e-'^^) or y(^) 3.}}}}}

5 : e"^(Di sin 6t + D2 cos bt) or y(^) = e^'^lEi sm{bt + ^2)] or y{t) = e''^[Ficos{bt + F2)] | Table : General Solution Forms for Second-Order Linear Homogeneous Equations , Constant Coefficients Homogeneous Equations of Order n When ( ) is of order n, the auxiliary equation p(m) = 0 has n roots, when multiple roots are coimted according to their multiplicity. Also, complex roots occur in conjugate pairs. The general Solutions of the homogeneous Equations is the sum of the Solutions associated with each multiple root. They can be foimd in Table for each root and should be added together to form the general solution.}}}

6 First, we give some examples in Table 366 A. Solutions of Linear Differential Equations Differential Equation 1. D2(2)2 _4D + 4)2/-0 2. (D-3)2(D + 5)3(D2-4D 3. (D2 -2D-\- 2)3(D2 - 2D -+ 5)2y=0 - 3)^2/ = 0 General Solution Form y(t) = Ci + C2i + e2*(C3 + C^t) 2/(t)-e3*(Ci+C2t) +e-5t(C3 + C4t + C5t2) +e2*[(C6+C7)sin t ^(Cs + CgOoos t] 2/(i) = e*[(Ci -f C2* + C3t2) sin t +(C4-f-C5t-f C6t2)cos i] +(C7 + Cst)e^^ + (Cg + Ciot)e-^ Table : Examples of Homogeneous Equations of Order n Root rrij, real Complex Conjugate aj lb bj2 Multiplicity r,=l rj >1 r,=l 0>1 General Solution Form yj{t) = Ce'^J^ yj{t) = (Ci 4- C2* +.}}

7 + Crjf-J -i)e"^i* e'^J*(Ci sin 6jt + C2 cos 6jt) e^i*[Ci + C2t + .. + Cr^-Ti -^)sin 6jt] +(Cr^- + l + Cr^+2t + .. + C2r,-r:'' -^) COS bjt] Table : General Solution Forms for Multiple Roots of Auxiliary Equation Particular Solutions of Linear with Constant Coefficients The particular solution to the inhomogeneous equation ( ) can usually be found by guessing the form of the answer and then verifying the guess by substitution. Table shows the correct forms for guessing for various kinds of forcing fimctions g(t). Note that the form of the guess depends on whether certain nimibers are roots of the auxiliary equation.

8 Table gives examples of Differential Equations along with their particular integrals. A,5. Particular Solutions of Linear D,E, Constant Coefficients 367 Forcing Function, g{t) (i)c (2) h{t) (3) csin qt or ccos qt (4) ce"* (5) ce^* sin qt or ce^* cos qt (6) /i(i)e * (7) /i(t) sin ^t or /i(f) cos qt (8) /i(t)eP*sin gt or /i(i)eP*cos gt K 0 0 9 9 p + iq Q iq p + iq Particular Integral, y{t) A X A sin qt B cos qt Ae<i* AeP* sin ^t - Be^* cos ^i Xe * X sin gi + y cos qfi XeP*sin ^i + FeP^cos qt Notation. (a) In the forcing function column, p, q, and c are given constants and h{t) is a given polynomial of degree s.}}}}

9 (b) In the p>articular integral column, A and B are coefficients to be determined and X = ^0 + Alt + .. + Ast\ Y = Bo-\-Bit+ .. + Bst^ are s degree polynomials whose coefficients are to be determined. Rules. (a) If the number in the K column is not a root of the auxiliary equation p(m) = 0, then the proper guess for the particular integral is as shown. (b) If the number in the K colimin is a root of the auxiliary equation of degree r, then multiply the guess in the last column by t^. Table : Particular Solution Forms for Various Forcing Functions If the forcing function g{t) is the sum of several functions, 9^=91 + g2 + *"+9ky each having one of the forms in the table, then solve for each Qi separately and add the results together to get the complete solution.}

10 In the next table, we wiU apply the formulas and the rules in Table to obtain particular integrals in specific examples. 368 A. Solutions of Linear Differential Equations Differential Equation '"-32/" = 5 2. y'" - 3y" = l + 3t + 5t^ 3. 2/"-V + 42/ = 3-i2 4. y" -4y' + 4y = 2sm t 5. 2/"-43/'+ 42/= 5sin 2t 6. 2/" - 42/' + 4y = lOe^* 7. 2/" - 4y' + 42/ = lOe^* 8. 2/" - 42/' + 52/ = e^* cos t 9. 2/"-42/' + 52/ = t^sin f 10. y" - 42/' + 5y = i^e^* cos t Particular Integral At^ t^Ao + Ait + A2t'^) Ao + Ait + A2t^ Asm t + Bcos t Asm 2t + Bcos 2t Ae'^ i2(Ae2 ) t{Ae^^ sin t + Be^* cos t) X^r_o(^r^'^ sin i + B^V cos i) * Er=o(^r*''e2* sin + B^fe^* cos *)J Table : Particular Integrals in Specific Examples Integrating Factor Consider the first-order Linear equation 2/' + ay = /(i).}