Differential Equations d'yFind the general solution to the ODEdt2dy+ 10y = 0. Enter solutions in alphabetical order.dty(t) = c1* +c2

The given equation is d2ydt2-2dydt+10y=0 The auxiliary equation is D2-2D+10y=0 Solve for D…

Answered: d'y dy 2- dt Find the general solution…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Differential Equations

d'y
Find the general solution to the ODE
dt2
dy
+ 10y = 0. Enter solutions in alphabetical order.
dt
y(t) = c1
* +c2

Expert Solution

Step 1

The given equation is $\frac{d^{2} y}{d t^{2}} - 2 \frac{d y}{d t} + 10 y = 0$

The auxiliary equation is $(D^{2} - 2 D + 10) y = 0$

Solve for D

$\begin{array}{rcl} D^{2} - 2 D + 10 & = & 0 \\ D & = & \frac{- (- 2) \pm \sqrt{{(- 2)}^{2} - 4 (1) (10)}}{2 (1)} \\ = & \frac{2 \pm \sqrt{4 - 40}}{2} \\ = & \frac{2 \pm \sqrt{- 36}}{2} \\ = & \frac{2 \pm 6 i}{2} \\ = & 1 \pm 3 i \end{array}$

The roots are imaginary of the form $α \pm β i$ therefore, the solution of the differential equation is given by

$y = e^{α t} (c_{1} \cos β t + i \sin β t)$

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