Given , `y=ae^(3x)+be^(-2x)` ………….`(1)`
`implies (dy)/(dx)=a(d)/(dx)(e^(3x))+b(d)/(dx)(e^(-2x))`
`implies (d)/(dy)/(dx)=ae^(3x)(d)/(dx)(3x)+be^(-2x)(d/(dx)(-2x)`
`implies (dy)/(dx)=3ae^(3x)-2be^(-2x)`……..`(2)`
`implies(d^(2)y)/(dx^(2))=3ae^(3x)(d)/(dx)(3x)-2be^(-2x)(d)/(dx)(-2x)`
`=9ae^(3x)+4be^(-2x)`
`implies (d^(2)y)/(dx^(2))=3(3ae^(3x))+4be^(-2x)`
`=9ae^(3x)+4be^(-2x)`...........`(3)`
Multiply equation `(2)` by `3` and subtracting from equation `(3)`,
`implies (d^(2)y)/(dx^(2))-3(dy)/(dx)=10be^(-2x)` ...........`(4)`
Multiply equation `(1)` by `3` and subtracting from equation `(2)`,
`(dy)/(dx)-3y=-2be^(-2x)-3be^(-2x)`
`implies 3y-(dy)/(dx)=5be^(-2x)`..........`(5)`
Divide equation `(4)` by `(5)`
`((d^(2)y)/(dx^(2))-(3dy)/(dx))/(3y-(dy)/(dx))=2`
`implies (d^(2)y)/(dx^(2))-3(dy)/(dx)=6y-(2dy)/(dx)`
`implies (d^(2)y)/(dx^(2))-(dy)/(dx)-6y=0`
which is required differential equation.
