solve-D-2-a-y-tan-ax-

Question Number 126683 by bramlexs22 last updated on 23/Dec/20

s o l v e (D^{2} + a) y = \tan a x

Answered by liberty last updated on 23/Dec/20

(1) H o m o g e n o u s s o l u t i o n

y_{h} = A \cos a x + B \sin a x

(2) P a r t i c u l a r s o l u t i o n

y_{1} = \cos a x \to y_{1}^{'} = - a \sin a x

y_{2} = \sin a x \to y_{2}^{'} = a \cos a x

W r o n s k i a n W (y_{1}, y_{2}) = | \begin{matrix} \cos a x \sin a x \\ - a \sin a x a \cos a x \end{matrix} | = a

p u t u = - \int \frac{y_{2} g (x)}{W} d x = - \int \frac{\sin a x \tan a x}{a} d x

= \frac{1}{a^{2}} (\sin a x - \ln ∣ \sec a x + \tan a x ∣)

p u t v = \int \frac{y_{1} g (x)}{W} d x = \int \frac{\cos a x \tan a x}{a} d x

= - \frac{\cos a x}{a^{2}}

y_{p} = {u y}_{1} + {v y}_{2} = - \frac{1}{a^{2}} \cos a x \ln ∣ \sec a x + \tan a x ∣

(3) y_{g} = A \cos a x + B \sin a x - \frac{\cos a x \ln ∣ \sec a x + \tan a x ∣}{a^{2}}

Answered by mathmax by abdo last updated on 24/Dec/20

y^{^{″}} + ay = tanx

h \to r^{2} + a = 0

case 1 a > 0 \Rightarrow r^{2} = - a \Rightarrow r = \bar{+} \sqrt{- a} = \bar{+} i \sqrt{a} \Rightarrow

y_{h} = {ae}^{i \sqrt{a} x} + {be}^{- i \sqrt{a} x} = α \cos (\sqrt{a} x) + β \sin (\sqrt{a} x) = α u_{1} + β u_{2}

w (u_{1}, u_{2}) = | \begin{matrix} \cos (\sqrt{a} x) \sin (\sqrt{a} x) \\ - \sqrt{a} \sin (\sqrt{a} x) \sqrt{a} \cos (\sqrt{a} x) \end{matrix} | = \sqrt{a}

w_{1} = | \begin{matrix} o \sin (\sqrt{a} x) \\ \tan (ax) \sqrt{a} \cos (\sqrt{a} x) \end{matrix} | = - \sin (\sqrt{a} x) \tan (ax))

w_{2} = | \begin{matrix} \cos (\sqrt{a} x) 0 \\ - \sqrt{a} \sin (\sqrt{a} x) \tan (ax) \end{matrix} | = \cos (\sqrt{a} x) \tan (ax)

v_{1} = \int \frac{w_{1}}{w} dx = - \int \frac{\sin (\sqrt{a} x) \tan (ax)}{\sqrt{a}} dx = - \frac{1}{\sqrt{a}} \int \sin (\sqrt{a} x) \tan (ax) dx

v_{2} = \int \frac{w_{2}}{w} dx = \int \frac{\cos (\sqrt{a} x) \tan (ax)}{\sqrt{a}} dx

\Rightarrow y_{p} = u_{1} v_{1} + u_{2} v_{2} = - \frac{1}{\sqrt{a}} \cos (\sqrt{a} x) \int \sin (\sqrt{a} x) \tan (ax) dx

+ \frac{1}{\sqrt{a}} \sin (\sqrt{a} x) \int \cos (\sqrt{a} x) \tan (ax) dx \Rightarrow

general solution is y = y_{p} + h_{h}

case 2 < 0 r^{2} + a = 0 \Rightarrow r^{2} = - a \Rightarrow r = \bar{+} \sqrt{a} \Rightarrow

y_{h} = a e^{\sqrt{a} x} + {be}^{- \sqrt{a} x} = {au}_{1} + {bu}_{2} and we follow the same way \dots