Question

2. Solve the following second order homogeneous differential equations: a) *+x+2x =0 b) Ö-70+5Q =0 c) y"-6y'+9y=0 d) y"+9y=0.

Answer 1

Explanation:

- dt2 REMARK: 1. da da da dt 2. Consider a and order Homogenous differential equation, I with constant coefficents - tay' tby =o (1) Then the auxillary equation is - m²+ am to=o. - (2) On solving the equation (2), 3-cases arises: - Case: - I - when roots are district and real - if le 8 lz are two roots then ditla. Then, general solution is- Y(x) = çelix teet2x where ca, ca are arbitarety] - constants. Case: 2 - when roots are real and repeated. 20 aadi=x2. (say) Then general solution is Yx) = (a + (₂x) ex case: 3 when roots are complex. ie t= dI iß Then, general solution is- Yox)= x [c.cosBx + C2 Sinßx]. (9) Griveng tác +2x=0 Auxillary eqn is m² tm +2=0 solving using discriminant m= -b + b²_uac 29 of i ti Hence the general solution is- where c. 1627 1 y(x) = e 2* [c, cosuit + C2 SinJ7 27 (are arbitarety constants 2

b) ä -78 +50=0 8 8 da s = do Now, the auxiliary equation is - m² fm +5=0 Solving the quadratic equation using difcriminant We get the roots as- m. 7 1 Jag 2 Then the general solution is - 0 (7 +529) at (7-029/2t Q (t) = c. + се Where ca, ca are arbitarty a constants] OD Y" - by'+9450. Y'adqY"412 Auxillary equation is m² -6m +9=0 =) m²-3m-3m +9=0 - (m. 3)2 =0 in m=3,3 are the roots. (real and repeated) Then general solution is. Y(x) = (a + Cat) esot (d) y" +gyco Auxillary equation is ) m = t3c [complex root] Then, general solution is- y(t) = a cos 3t ₂ Sin3t I where 4, C2 are arbitarty constants]