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Find a general solution of the given differential equation. d²y dy 2-+ y = e'tan-1t dt2 dt

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section11.4: Plane Curves And Parametric Equations

Problem 33E

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Question

Find a general solution of the given
differential equation.
d²y
dy
- 2 +y= e'tan-1t
dt2
dt

Expert Solution

Step 1

Given differential equation is

$\frac{d^{2} y}{d t^{2}} - 2 \frac{d y}{d t} + y = e^{t} \tan^{- 1} t$

Associated homogeneous differential equation is

$\frac{d^{2} y}{d t^{2}} - 2 \frac{d y}{d t} + y = 0$

Auxiliary equation is

$m^{2} - 2 m + 1 = 0$

$\Rightarrow {(m - 1)}^{2} = 0$

$\Rightarrow m = 1, 1$

Roots of auxiliary are real and repeated.

So, the complementary solution is

$y_{c} = c_{1} e^{t} + c_{2} t e^{t}$ where $c_{1}$ and $c_{2}$ are arbitrary constants.

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