is where 21,2 are the solutions to the auxiliary equation ay" + by' + cy=0 y = A₁e¹¹x + A₂¹2x aλ² + bλ + c = 0. In the case of complex roots λ = piw, we instead used the solution y = ex (C cos wx + D sin @x) (a) To show that the two solutions are equivalent, substitute the complex roots into the first expression, and compare it to the second, identifying formulae for C and D in terms of A₁ and A2. (b) Now find formulae for A₁ and A₂ in terms of C and D. If C and D are real (so that the solution is real), what is the relation between A₁ and A₂?
is where 21,2 are the solutions to the auxiliary equation ay" + by' + cy=0 y = A₁e¹¹x + A₂¹2x aλ² + bλ + c = 0. In the case of complex roots λ = piw, we instead used the solution y = ex (C cos wx + D sin @x) (a) To show that the two solutions are equivalent, substitute the complex roots into the first expression, and compare it to the second, identifying formulae for C and D in terms of A₁ and A2. (b) Now find formulae for A₁ and A₂ in terms of C and D. If C and D are real (so that the solution is real), what is the relation between A₁ and A₂?
the the solution of a 2nd order linear ODE with constant coefficients (see picture for rest of question)
Transcribed Image Text:is
where 1,2 are the solutions to the auxiliary equation
ay" + by' + cy=0
y = A₁e¹¹x + A₂¹²x
aλ² + bλ + c = 0.
In the case of complex roots λ = piw, we instead used the solution
y = ex (C cos @x + D sin @x)
(a) To show that the two solutions are equivalent, substitute the complex roots into the first expression, and compare it to the second,
identifying formulae for C and D in terms of A₁ and A₂.
(b) Now find formulae for A₁ and A₂ in terms of C and D. If C and D are real (so that the solution is real), what is the relation between A₁
and A2?
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