Solve the following linear second order homogeneous differential equationy&nbsp;″−32y′+916y=0,&nbsp;y(0)=1,&nbsp;y'(0)=-1

Solution:y″−32y′+916y=0, y(0), y′(0)=−1The characteristic equation is given byγ2−32γ+916=0⇒16γ2−24γ+9=0Henceγ=24±242−4⋅16⋅92⋅16=24±576−57632=2432=34Since the root are repeated therefore the solution arey1(t)=eγ⋅t and y2(t)=teγ⋅t=e34 and y2(t)=te34⋅tLet yc be the complimentary soltThenyc(t)=c1y1+c2y2, where c1 and c2 constant=c1e34⋅t+c2te34⋅t=e34⋅t(c1+c2⋅t)given, y(0)=1, soy(0)=1 c1=1y′(t)=34e34t(c1+c2t)+e34tc2y′(0)=34c1+c2=−1c2=−1−34c1=−1−34=−74Hencey(t)=e34t−74te34t=e34(1−74t)

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Blackettyl2j

Answered question

2022-03-25

Solve the following linear second order homogeneous differential equation
$y^{″} - \frac{3}{2} y^{'} + \frac{9}{16} y = 0,$ $y (0) = 1, y' (0) = - 1$

Answer & Explanation

Gia Edwards

Beginner2022-03-26Added 12 answers

Solution:

y^{″} - \frac{3}{2} y^{'} + \frac{9}{16} y = 0, y (0), y^{'} (0) = - 1

The characteristic equation is given by

γ^{2} - \frac{3}{2} γ + \frac{9}{16} = 0

\Rightarrow 16 γ^{2} - 24 γ + 9 = 0

Hence

γ = \frac{24 \pm \sqrt{24^{2} - 4 \cdot 16 \cdot 9}}{2 \cdot 16}

= \frac{24 \pm \sqrt{576 - 576}}{32}

= \frac{24}{32} = \frac{3}{4}

Since the root are repeated therefore the solution are

y_{1} (t) = e^{γ \cdot t}

and

y_{2} (t) = t e^{γ \cdot t}

= e^{\frac{3}{4}}

and

y_{2} (t) = t e^{\frac{3}{4} \cdot t}

Let

y_{c}

be the complimentary solt
Then

y_{c} (t) = c_{1} y_{1} + c_{2} y_{2}

, where

c_{1}

and

c_{2}

constant

= c_{1} e^{\frac{3}{4} \cdot t} + c_{2} t e^{\frac{3}{4} \cdot t}

= e^{\frac{3}{4} \cdot t} (c_{1} + c_{2} \cdot t)

given,

y (0) = 1

, so

y (0) = 1 c_{1} = 1

y^{'} (t) = \frac{3}{4} e^{\frac{3}{4} t} (c_{1} + c_{2} t) + e^{\frac{3}{4} t} c_{2}

y^{'} (0) = \frac{3}{4} c_{1} + c_{2} = - 1

c_{2} = - 1 - \frac{3}{4} c_{1}

= - 1 - \frac{3}{4} = - \frac{7}{4}

Hence

y (t) = e^{\frac{3}{4} t} - \frac{7}{4} t e^{\frac{3}{4} t}

= e^{\frac{3}{4}} (1 - \frac{7}{4} t)

Jeffrey Jordon

Expert2022-03-31Added 2605 answers

Answer is given below (on video)

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