#### 2.23   ODE No. 23

$a y(x)^2-b+y'(x)=0$ Mathematica : cpu = 0.062718 (sec), leaf count = 43

$\left \{\left \{y(x)\to \frac {\sqrt {b} \tanh \left (\sqrt {a} \sqrt {b} c_1+\sqrt {a} \sqrt {b} x\right )}{\sqrt {a}}\right \}\right \}$ Maple : cpu = 0.033 (sec), leaf count = 23

$\left \{ y \left ( x \right ) ={\frac {1}{a}\tanh \left ( \sqrt {ab} \left ( x+{\it \_C1} \right ) \right ) \sqrt {ab}} \right \}$

Hand solution

\begin {align*} y^{\prime }+ay^{2}-b & =0\\ \frac {dy}{dx} & =b-ay^{2} \end {align*}

Separable,\begin {align*} \frac {dy}{b-ay^{2}} & =dx\\ \int \frac {dy}{b-ay^{2}} & =\int dx \end {align*}

But $\int \frac {dy}{b-ay^{2}}=\frac {1}{\sqrt {ab}}\tanh ^{-1}\left ( \sqrt {\frac {a}{b}}y\right )$ Hence\begin {align*} \frac {1}{\sqrt {ab}}\tanh ^{-1}\left ( \sqrt {\frac {a}{b}}y\right ) & =x+C\\ \tanh ^{-1}\left ( \sqrt {\frac {a}{b}}y\right ) & =\sqrt {ab}\left ( x+C\right ) \\ \sqrt {\frac {a}{b}}y & =\tanh \left ( \sqrt {ab}\left ( x+C\right ) \right ) \\ y & =\sqrt {\frac {b}{a}}\tanh \left ( \sqrt {ab}\left ( x+C\right ) \right ) \end {align*}