# Thread: I need some help with 3 prob.s from my AP class...

1. ## I need some help with 3 prob.s from my AP class...

Solve the following using separation of variables. Let C represent an arbitrary constant.

1) (11 + x^2)y ' = 11x^(3)*y

^ I got (Ce^((11x^2)/2))/(e^(11log(x^2+11))) but that's not correct.

Solve the initial value problem.

2) y ' - 2y + 4 = 0, y(1) = 12

^I don't even know how to approach this one.

3)y^2 * dy/dx = x^(-3), y(5) = 0

^I got ((25-x)/6)^(1/3) but that's wrong.

I'm not necessarily looking for the answers...but take me step my step? Or atleast guide me through one or two of them?

2. Hello, woohoo!

$1)\;\;(11 + x^2)\,\frac{dy}{dx}\:=\: 11x^3y$

Separate variables: . $\frac{dy}{y} \;=\;11\,\frac{x^3}{x^2+11} \quad\Rightarrow\quad \frac{dy}{y} \;=\;11\left(x - \frac{11x}{x^2+11}\right)$

Integrate: . $\int\frac{dy}{y} \;=\;11\left[\int x\,dx - 11\int\frac{x\,dx}{x^2+11}\right]$

Therefore: . $\ln y \;=\;11\bigg[\frac{x^2}{2} - \frac{11}{2}\ln(x^2+11)\bigg] + C$

$2)\;\;\frac{dy}{dx} - 2y + 4 \:=\: 0,\quad y(1) = 12$
We have: . $\frac{dy}{dx} - 2y \:=\:-4$

Integrating factor: . $I \:=\:e^{\int(-2)\,dx} \:=\:e^{-2x}$

Multiply by $I\!:\;\;e^{-2x}\,\frac{dy}{dx} - 2e^{-2x}y \;=\;-4e^{-2x} \quad\Rightarrow\quad \frac{d}{dx}\left(e^{-2x}y\right) \;=\;-4e^{-2x}$

Integrate: . $e^{-2x}y \;=\;2e^{-2x} + C \quad\Rightarrow\quad y \;=\;2 + Ce^{2x}$

Since $y(1) = 12$, we have: . $12 \;=\;2 + Ce^2 \quad\Rightarrow\quad C \:=\:\frac{10}{e^2}$

Therefore: . $y \;=\;2 + \frac{10}{e^2}\,e^{2x} \quad\Rightarrow\quad y \;=\;2 + 10e^{2x-2}$

$3)\;\;y^2\,\frac{dy}{dx} \:=\:x^{-3},\quad y(5) = 0$

Separate variables: . $y^2dy \;=\;x^{-3}dx$

Integrate: . $\frac{y^3}{3} \;=\;\frac{x^{-2}}{-2} + C \quad\Rightarrow\quad y^3 \;=\;-\frac{3}{2x^2} +C$

Since $y(5) = 0$, we have: . $0^3 \;=\;-\frac{3}{2(5^2)} + C \quad\Rightarrow\quad C \:=\:\frac{3}{50}$

Therefore: . $y^3 \;=\;-\frac{3}{2x^2} + \frac{3}{50}$

3. Sweet!

Now, if i were to multiply both sides of the first one by e, I would get...

y=Ce^(11x^2)/e^(11ln(x^2+11)/2) correct?

how can I simplify this?

would it be:

2Ce^((11x^2)/2)/(11x^2+121) ?