Yup, still doing DE!

Calculus Level 4

\dddoty2y¨5y˙+6y=0\large \dddot y - 2 \ddot y - 5 \dot y +6y=0

If the above differential equation has the general solution of the form y=a1eat+a2ebt+a3ecty = { a }_{ 1 }{ e }^{ at }+{ a }_{ 2 }{ e }^{ bt }+{ a }_{ 3 }{ e }^{ ct }, where a1 a_1, a2a_2 and a3a_3 are real constants, and aa, bb and cc are integers, find a+b+c a+b+c.

Notations: y˙=dydt \dot y =\dfrac { dy }{ dt }, y¨=d2ydt2\ddot y =\dfrac { { d }^{ 2 }y }{ d{ t }^{ 2 } }, \dddoty=d3ydt3 \dddot y =\dfrac { { d }^{ 3 }y }{ d{ t }^{ 3 } } .

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