#### 2.97   ODE No. 97

$a y(x)^2+b x^2+x y'(x)-y(x)=0$ Mathematica : cpu = 0.100914 (sec), leaf count = 46

$\left \{\left \{y(x)\to -\frac {\sqrt {b} x \tan \left (\sqrt {a} \sqrt {b} x-\sqrt {a} \sqrt {b} c_1\right )}{\sqrt {a}}\right \}\right \}$ Maple : cpu = 0.032 (sec), leaf count = 25

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

Hand solution

$ay^{2}+bx^{2}+xy^{\prime }-y=0$

This is Riccati ﬁrst order non-linear. Let $$y=ux$$, hence the above becomes\begin {align*} au^{2}x^{2}+bx^{2}+x\left ( u^{\prime }x+u\right ) -ux & =0\\ au^{2}x+bx+u^{\prime }x & =0\\ au^{2}+b+u^{\prime } & =0\\ u^{\prime } & =-au^{2}-b \end {align*}

Which is separable, Hence

$\frac {du}{au^{2}+b}=-dx$

Integrating

\begin {align*} \frac {1}{\sqrt {ab}}\arctan \left ( \frac {au}{\sqrt {ab}}\right ) & =-x+C\\ \frac {au}{\sqrt {ab}} & =\tan \left ( \sqrt {ab}\left ( -x+C\right ) \right ) \\ u & =\frac {\sqrt {ab}}{a}\tan \left ( \sqrt {ab}\left ( -x+C\right ) \right ) \end {align*}

Therefore

\begin {align*} y & =ux\\ & =x\frac {\sqrt {ab}}{a}\tan \left ( \sqrt {ab}\left ( -x+C\right ) \right ) \end {align*}

Veriﬁcation

restart;
ode:=a*y(x)^2+b*x^2+x*diff(y(x),x)-y(x)=0;
my_sol:=x*sqrt(a*b)/a*tan(sqrt(a*b)*(-x+_C1));
odetest(y(x)=my_sol,ode);
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