How Do You Solve Linear Differential Equations with an Integrating Factor?

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methord to use?

find the solution of the initial value problem y'+(3/x)y=e2x/x3 y(1)=1
 
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What type of question do you think it is?
 
This is a linear differential equation. Probably the simplest way to do it is to find an integrating factor. Can you find a function, \mu(x) such that
\frac{d\mu y}{dx}= \mu\frac{dy}{dx}+\frac{d\mu}{dx}y= \mu\frac{dy}{dx}+(3\mu /x^3)y?
 

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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