Question

Find and classify the differential equation by eliminating a,b and c from general solution equation : y=ae^x+be^2x+ce^-3+x²

where a ,b ,and c are arbitrary constants

Answer 1

Given. y = ae" tbe tee ²x2 N differentiating with respect ton, y' = al' + 2be"+21-0 y"= ae'+4be?4+2 (again differentiating with reopeet to) Y" = aet a bet2be 442 " Y" =(y-2) +2be"+2 [using og] y"_y'+ 20-2= eben ērn (y_y'+20-2) = 20 again differentiating with respect to a -22** (79_y'+24-2)+ė (y"_y"+2) = 0 – 2y"+2y40+4 +y'"'- y" +2=0 ( ē22 +0) ly"-39"+ 2y =40-6 This is the required differential equation, linear and order differential equation with co-efficient . This is a constant

If the Given equation is, y =ae +be+cē3*+x2 y' = ae* + 2be?"- 3cē3*+ en Y" = a + 4be² +9ce342 Y" = ae" +8 bez _ 27 00 34 Re Re-writting the above equations. 3 -34 yox² = ae + bent een y-2x = ae* +2b8%a_zcē 3u Y"-2 =ae" +4be9ce34 = ala+86e2_27cé Eleminating a,b,c from 0 20and 4 y". we get. си e e 34 =0 7. 22 en 221 -303 en 4e2n Y"-2 yt 90 34 Pela -27 034) 1 1 Taking common from and ard and 4th 2 -3 a e? 1 2 3 4 022 34 /y-x? y-2x y"-2 4 9 column] 1 8 -27 112-30 435 1831-(4- ********* [expanding with resp. N to get column] -120(y-u?) – (140) (y-2) +0. 5. (y-2) – 207" = 0 or – 207" +1404²-1204 = -120x272800 201" - 1400'+1204 Ty" -7y'+by = 6x2 142 J2004-2008