Question

2. Use the method for solving homogeneous equation to solve the following differential equation (6y2 – xy)dx + x?dy = 0 3. Find a general solution to the given differential equation 49w" + 140w' + 100w = 0

Answer 1

(2) (6 4²-xy) dat &? dy = 0 or (642-xy) dx = -x?dy 642-xy dx dy dy da 0 dy xy-642 x² Now Put yaux dy - +xdu then becomes ax da O+xdu = x(2x), 6 (Vx)? da x2 vx²-6u²x² x2 X? (L-602) x² 11 vtx du da 2 و61 - سوا xdu dx v-60² v redu - 6 us 2 da 1 du - 2 seprating variables. - Integrate on both side, we will get Sea = 6) 1 de -6) = 6inx + c 6enate du ells 2 = 6 enate ( + - x) y 1 6 enate V y = x Genate (Result)

Quven 49w" + 140 w'+ 100 W=0 characterstic equation of the following differential equation can be written as 49 m2 + 140 m+ 100 = 0 49 m2 + (70+70) m + 10020 49m² + tom & tomt 2000 Fm (7m + 10) + 10 (7m + 10) - 0 (7m+ 10) (7m+ 10) - the roots are 7 Now for two real and repeated root the general solution is given by W(t) = (6 + C₂ t) em: Hence me -40 10 7 9 m mit Hence W(+) = (61+ (at) e 20 out general solution of the differential This is the required equation.