Internal problem ID [3653]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 14
Problem number: 399.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Riccati, _special]]
\[ \boxed {x^{\frac {3}{2}} y^{\prime }-b \,x^{\frac {3}{2}} y^{2}=a} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 122
dsolve(x^(3/2)*diff(y(x),x) = a+b*x^(3/2)*y(x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {2 a \left (\operatorname {BesselJ}\left (1, 4 \sqrt {a}\, \sqrt {b}\, x^{\frac {1}{4}}\right ) c_{1} +\operatorname {BesselY}\left (1, 4 \sqrt {a}\, \sqrt {b}\, x^{\frac {1}{4}}\right )\right )}{\sqrt {x}\, \left (2 \sqrt {a}\, \operatorname {BesselJ}\left (0, 4 \sqrt {a}\, \sqrt {b}\, x^{\frac {1}{4}}\right ) x^{\frac {1}{4}} \sqrt {b}\, c_{1} +2 \operatorname {BesselY}\left (0, 4 \sqrt {a}\, \sqrt {b}\, x^{\frac {1}{4}}\right ) \sqrt {a}\, \sqrt {b}\, x^{\frac {1}{4}}-\operatorname {BesselJ}\left (1, 4 \sqrt {a}\, \sqrt {b}\, x^{\frac {1}{4}}\right ) c_{1} -\operatorname {BesselY}\left (1, 4 \sqrt {a}\, \sqrt {b}\, x^{\frac {1}{4}}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.252 (sec). Leaf size: 373
DSolve[x^(3/2) y'[x]==a+ b x^(3/2) y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {a} \sqrt {b} \sqrt [4]{x} \operatorname {BesselY}\left (1,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )+\operatorname {BesselY}\left (2,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )-\sqrt {a} \sqrt {b} \sqrt [4]{x} \operatorname {BesselY}\left (3,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )-\sqrt {a} \sqrt {b} c_1 \sqrt [4]{x} \operatorname {BesselJ}\left (1,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )-c_1 \operatorname {BesselJ}\left (2,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )+\sqrt {a} \sqrt {b} c_1 \sqrt [4]{x} \operatorname {BesselJ}\left (3,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )}{2 b x \operatorname {BesselY}\left (2,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )-2 b c_1 x \operatorname {BesselJ}\left (2,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )} y(x)\to -\frac {\sqrt {a} \sqrt {b} \sqrt [4]{x} \operatorname {BesselJ}\left (1,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )+\operatorname {BesselJ}\left (2,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )-\sqrt {a} \sqrt {b} \sqrt [4]{x} \operatorname {BesselJ}\left (3,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )}{2 b x \operatorname {BesselJ}\left (2,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )} \end{align*}