Get Expert Help with Homogeneous Equations and Newton's Law of Cooling

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hey guys i need help with this problem i tried to figure the out but most are homogeneous equations so i don't even know how to start and the Newtons law of cooling i have no idea wut to do .thanks
 

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Welcome to the forums aparra2. Firstly, please note that all homework/textbook questions like this should be posted in the homework forums. Also note that you need to show your work before we can help you- forum rules.

For this question, your image is not clear, and I can't quite read the questions. Perhaps you could type the equations out? If you do, then I may be able to give you hints; but are you sure you don't know where to start? What do you know about how to solve these type of equations?
 
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Is the first equation
p\frac{dp}{dt}= 1+ x+ 2y+ 2xy?
If so, are we to treat y as a constant?

Is the second equation
(1+ x^2)\frac{dp}{dx}+ 4xp= \frac{1}{1+x^2}?
If so, that is a linear equation. There is a standard formula for finding an "integrating factor". Do you know it?

Is the third equation
(p+ t^2y)\frac{dy}{dx}= 2tdt?
If so can we treat p as a constant?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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