00:01
In this problem we have to find the general solution to the given differential equation.
00:06
So the problem a is here, the differential equation 4 y double dash plus 11 y -dice minus 3y equal 0.
00:16
So to find the general solution, we will write here the auxiliary equation.
00:24
So auxiliary equation is 4r square plus 11 r minus 3 equal 0.
00:33
Now we will factor this equation.
00:35
So 4r squared plus 12r minus r minus 3 equal 0.
00:41
So taking common here 4.
00:43
So 4 r plus 3 minus 1 r plus 3 equal 0.
00:50
So taking common r plus 3 we have 4 r minus 1 equal 0.
00:57
So r plus 3 equal 0 that means r equal minus 3.
01:02
So this is r1 equal minus 3 let's take this as r1 now 4r minus 1 equal 0 this implies r equal 1 by 4 so let's take it as r2 equal 1 by 4.
01:20
Now if we have two distinct real roots r1 and r2 then the general solution of this differential equation is y x equal c1 to power r1 x plus c2 e to power r2 x so putting this value c1 e to power r1 is minus 3x plus c2 e to power 1 by 4 x this is the general solution for the problem number a where this c1 and c2 are the arbitrary constants so this c1 and c2 are arbitrary constant now we will find the solution for for problem b...