- #1
lol_nl
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Homework Statement
Out of a set of differential equations with boundary conditions, there are three (first order) equations I couldn't solve. These are:
Homework 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]
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.