29.7.5 problem 180

Internal problem ID [4780]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 7
Problem number : 180
Date solved : Monday, January 27, 2025 at 09:37:35 AM
CAS classification : [_rational, _Riccati]

\begin{align*} x y^{\prime }+a \,x^{2} y^{2}+2 y&=b \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 68

dsolve(x*diff(y(x),x)+a*x^2*y(x)^2+2*y(x) = b,y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {\left (-\operatorname {BesselY}\left (1, \sqrt {-a b}\, x \right ) c_{1} -\operatorname {BesselJ}\left (1, \sqrt {-a b}\, x \right )\right ) \sqrt {-a b}}{a x \left (c_{1} \operatorname {BesselY}\left (0, \sqrt {-a b}\, x \right )+\operatorname {BesselJ}\left (0, \sqrt {-a b}\, x \right )\right )} \]

✓ Solution by Mathematica

Time used: 0.237 (sec). Leaf size: 158

DSolve[x D[y[x],x]+a x^2 y[x]^2+2 y[x]==b,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {i \sqrt {b} \left (\operatorname {BesselY}\left (1,-i \sqrt {a} \sqrt {b} x\right )-c_1 \operatorname {BesselJ}\left (1,i \sqrt {a} \sqrt {b} x\right )\right )}{\sqrt {a} x \left (\operatorname {BesselY}\left (0,-i \sqrt {a} \sqrt {b} x\right )+c_1 \operatorname {BesselJ}\left (0,i \sqrt {a} \sqrt {b} x\right )\right )} \\ y(x)\to -\frac {i \sqrt {b} \operatorname {BesselJ}\left (1,i \sqrt {a} \sqrt {b} x\right )}{\sqrt {a} x \operatorname {BesselJ}\left (0,i \sqrt {a} \sqrt {b} x\right )} \\ \end{align*}