(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Out of a set of differential equations with boundary conditions, there are three (first order) equations I couldn't solve. These are:

2. Relevant equations

1. [tex]

\frac {dy} {dx} = \sqrt{x + y}, y(1) = 0.

[/tex]

2. [tex]

\frac {dy} {dx} = 2y(x \sqrt{y} - 1), y(0) = 1.

[/tex]

3. [tex]

2x^2 \frac {dy} {dx} = x^2 + y^2, y(2) = 4.

[/tex]

3. The attempt at a solution

The first two can probably be solved with a nice substitution. I tried u = x + y for the first one, but this gave me the equation [tex]\frac {du} {dx} = 1 + \sqrt{u}[/tex], which can be solved for u to get, after resubstituting, [tex]2 \sqrt{x+y} - 2 log[1+\sqrt{x+y} - 2 + 2 log[2] = x[/tex], which doesn't seem solvable for y[x].

For the second one I tried substituting [tex]u = x \sqrt{y} [/tex] and [tex]u = x \sqrt{y} - 1 [/tex], but neither gave an equation that could be written in terms of u only (without x or y).

For the third one I only noticed that y=x is a general solution, but it doesn't agree with the initial condition y(2) = 4, and neither does any manipulation such as y=2x or y=x+2.

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# Homework Help: Three linear first-order ODEs

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