MHB -b.2.2.18 IVP DE complete the square?

karush
Gold Member
MHB
Messages
3,240
Reaction score
5
$\quad\displaystyle
y^{\prime}=
\frac{e^{-x}-e^x}{3+4y},
\quad y(0)=1$
rewrite
$\frac{dy}{dx}=\frac{e^{-x}-e^x}{3+4y}$
separate
$3+4y \, dy = e^{-x}-e^x \, dx$
integrate
$2y^2+3y=-e^{-x}-e^x+c$
well if so far ok presume complete the square ?book answer
$(a)\quad y=-\frac{3}{4}+\frac{1}{4}
+\sqrt{65-8e^x-8e^{-x}}$\\
$(c)\quad|x|<2.0794\textit{ approximately}$
 
Last edited:
Physics news on Phys.org
karush said:
$\quad\displaystyle
y^{\prime}=
\frac{e^{-x}-e^x}{3+4y},
\quad y(0)=1$
rewrite
$\frac{dy}{dx}=\frac{e^{-x}-e^x}{3+4y}$
separate
$3+4y \, dy = e^{-x}-e^x \, dx$
integrate
$2y^2+3y=-e^{-x}-e^x+c$
well if so far ok presume complete the square ?book answer
$(a)\quad y=-\frac{3}{4}+\frac{1}{4}
+\sqrt{65-8e^x-8e^{-x}}$\\
$(c)\quad|x|<2.0794\textit{ approximately}$

$y(0) = 1 \implies 5 = -2+c \implies c = 7$

$y^2 + \dfrac{3}{2}y = \dfrac{-e^{-x} -e^x + 7}{2}$

$y^2 + \dfrac{3}{2}y + \dfrac{9}{16} = \dfrac{-e^{-x} -e^x + 7}{2} + \dfrac{9}{16} $

$\left(y+\dfrac{3}{4}\right)^2 = \dfrac{-8e^{-x}-8e^x+65}{16}$

$y+\dfrac{3}{4} = \dfrac{\sqrt{-8e^{-x}-8e^x+65}}{4}$

check the book "answer" ...

$y = \dfrac{-3 + \sqrt{-8e^{-x}-8e^x+65}}{4}$
 
book answer

View attachment 8681

thanks for all the steps

i get lost on the initial value thing

I don't see where the 5 comes from?
 

Attachments

  • 18.png
    18.png
    3.7 KB · Views: 124
Last edited:
karush said:
book answer
thanks for all the steps

i get lost on the initial value thing

I don't see where the 5 comes from?

$2y^2+3y=-e^{-x}-e^x+C$

note $y(0)=1 \implies y = 1 \, \text{when} \, x=0$

$2(1)^2 + 3(1) = -e^{-0}-e^0+C$

... see it now?
 
Thread 'Direction Fields and Isoclines'
I sketched the isoclines for $$ m=-1,0,1,2 $$. Since both $$ \frac{dy}{dx} $$ and $$ D_{y} \frac{dy}{dx} $$ are continuous on the square region R defined by $$ -4\leq x \leq 4, -4 \leq y \leq 4 $$ the existence and uniqueness theorem guarantees that if we pick a point in the interior that lies on an isocline there will be a unique differentiable function (solution) passing through that point. I understand that a solution exists but I unsure how to actually sketch it. For example, consider a...

Similar threads

Replies
5
Views
1K
Replies
5
Views
2K
Replies
3
Views
1K
Replies
4
Views
2K
Replies
5
Views
2K
Replies
2
Views
1K
Replies
9
Views
2K
Back
Top