#### 2.100   ODE No. 100

$a+x y'(x)+x y(x)^2=0$ Mathematica : cpu = 0.0755735 (sec), leaf count = 157

$\left \{\left \{y(x)\to -\frac {c_1 J_1\left (2 i \sqrt {-a} \sqrt {x}\right )+i \sqrt {-a} \sqrt {x} \left (c_1 J_0\left (2 i \sqrt {-a} \sqrt {x}\right )-c_1 J_2\left (2 i \sqrt {-a} \sqrt {x}\right )-2 J_0\left (2 i \sqrt {-a} \sqrt {x}\right )\right )}{2 x \left (J_1\left (2 i \sqrt {-a} \sqrt {x}\right )-c_1 J_1\left (2 i \sqrt {-a} \sqrt {x}\right )\right )}\right \}\right \}$ Maple : cpu = 0.056 (sec), leaf count = 59

$\left \{ y \left ( x \right ) ={1\sqrt {a} \left ( {{\sl J}_{0}\left (2\,\sqrt {a}\sqrt {x}\right )}{\it \_C1}+{{\sl Y}_{0}\left (2\,\sqrt {a}\sqrt {x}\right )} \right ) {\frac {1}{\sqrt {x}}} \left ( {\it \_C1}\,{{\sl J}_{1}\left (2\,\sqrt {a}\sqrt {x}\right )}+{{\sl Y}_{1}\left (2\,\sqrt {a}\sqrt {x}\right )} \right ) ^{-1}} \right \}$

Hand solution

\begin {align*} xy^{\prime }+xy^{2}+a & =0\\ y^{\prime } & =-\frac {a}{x}-y^{2} \end {align*}

This is Riccati ﬁrst order non-linear. Let $$y=-\frac {u^{\prime }}{uR}=\frac {u^{\prime }}{u}$$, hence $$y^{\prime }=\frac {u^{\prime \prime }}{u}-\frac {\left ( u^{\prime }\right ) ^{2}}{u^{2}}$$. Equating this to RHS of the above gives\begin {align*} \frac {u^{\prime \prime }}{u}-\frac {\left ( u^{\prime }\right ) ^{2}}{u^{2}} & =-\frac {a}{x}-\left ( \frac {u^{\prime }}{u}\right ) ^{2}\\ \frac {u^{\prime \prime }}{u} & =-\frac {a}{x}\\ u^{\prime \prime }+\frac {a}{x}u & =0 \end {align*}

This is linear second order, an Emden Fowler ODE, with removal singularity. Solved using power series method. The solution is$u=C_{1}\sqrt {x}\operatorname {BesselJ}\left ( 1,2\sqrt {ax}\right ) +C_{2}\sqrt {x}\operatorname {BesselY}\left ( 1,2\sqrt {ax}\right )$ But $\frac {d}{dx}\operatorname {BesselJ}\left ( 1,2\sqrt {ax}\right ) =\frac {\sqrt {a}}{\sqrt {x}}\left ( \operatorname {BesselJ}\left ( 0,2\sqrt {ax}\right ) -\frac {1}{2}\frac {1}{\sqrt {ax}}\operatorname {BesselJ}\left ( 1,2\sqrt {ax}\right ) \right )$ And$\frac {d}{dx}\operatorname {BesselY}\left ( 1,2\sqrt {ax}\right ) =\frac {\sqrt {a}}{\sqrt {x}}\left ( \operatorname {BesselY}\left ( 0,2\sqrt {ax}\right ) -\frac {1}{2}\frac {1}{\sqrt {ax}}\operatorname {BesselY}\left ( 1,2\sqrt {ax}\right ) \right )$ Therefore, \begin {align*} u^{\prime } & =C_{1}\left ( \frac {1}{2\sqrt {x}}\operatorname {BesselJ}\left ( 1,2\sqrt {a}\sqrt {x}\right ) +\sqrt {x}\frac {\sqrt {a}}{\sqrt {x}}\left ( \operatorname {BesselJ}\left ( 0,2\sqrt {a}\sqrt {x}\right ) -\frac {1}{2}\frac {1}{\sqrt {ax}}\operatorname {BesselJ}\left ( 1,2\sqrt {a}\sqrt {x}\right ) \right ) \right ) \\ & +C_{2}\left ( \frac {1}{2\sqrt {x}}\operatorname {BesselY}\left ( 1,2\sqrt {a}\sqrt {x}\right ) +\sqrt {x}\frac {\sqrt {a}}{\sqrt {x}}\left ( \operatorname {BesselY}\left ( 0,2\sqrt {ax}\right ) -\frac {1}{2}\frac {1}{\sqrt {ax}}\operatorname {BesselY}\left ( 1,2\sqrt {a}\sqrt {x}\right ) \right ) \right ) \end {align*}

Which is simpliﬁed to$u^{\prime }=C_{1}\sqrt {a}\operatorname {BesselJ}\left ( 0,2\sqrt {a}\sqrt {x}\right ) +C_{2}\sqrt {a}\operatorname {BesselY}\left ( 0,2\sqrt {a}\sqrt {x}\right )$ Therefore, from $$y=\frac {u^{\prime }}{u}$$, the solution is$y=\frac {C_{1}\sqrt {a}\operatorname {BesselJ}\left ( 0,2\sqrt {a}\sqrt {x}\right ) +C_{2}\sqrt {a}\operatorname {BesselY}\left ( 0,2\sqrt {a}\sqrt {x}\right ) }{C_{1}\sqrt {x}\operatorname {BesselJ}\left ( 1,2\sqrt {a}\sqrt {x}\right ) +C_{2}\sqrt {x}\operatorname {BesselY}\left ( 1,2\sqrt {a}\sqrt {x}\right ) }$ Let $$C=\frac {C_{1}}{C_{2}}$$, hence$y=\frac {C\sqrt {a}\operatorname {BesselJ}\left ( 0,2\sqrt {a}\sqrt {x}\right ) +\ \sqrt {a}\operatorname {BesselY}\left ( 0,2\sqrt {a}\sqrt {x}\right ) }{C\sqrt {x}\operatorname {BesselJ}\left ( 1,2\sqrt {a}\sqrt {x}\right ) +\ \sqrt {x}\operatorname {BesselY}\left ( 1,2\sqrt {a}\sqrt {x}\right ) }$

Veriﬁcation

restart;
ode:=x*diff(y(x),x)+x*y(x)^2+a=0;
num:=_C1*sqrt(a)*BesselJ(0,2*sqrt(a)*sqrt(x))+sqrt(a)*BesselY(0,2*sqrt(a)*sqrt(x));
den:=_C1*sqrt(x)*BesselJ(1,2*sqrt(a)*sqrt(x))+sqrt(x)*BesselY(1,2*sqrt(a)*sqrt(x));
my_solution:=num/den;
odetest(y(x)=my_solution,ode);
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