Internal problem ID [10811]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.3. Abel Equations of the Second Kind. subsection 1.3.3-2. Equations
of the form \(y y'=f_1(x) y+f_0(x)\)
Problem number: 74.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Abel, `2nd type`, `class A`]]
\[ \boxed {y y^{\prime }+a \left (1+2 b \sqrt {x}\right ) {\mathrm e}^{2 b \sqrt {x}} y=-a^{2} b \,x^{\frac {3}{2}} {\mathrm e}^{4 b \sqrt {x}}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 307
dsolve(y(x)*diff(y(x),x)+a*(1+2*b*x^(1/2))*exp(2*b*x^(1/2))*y(x)=-a^2*b*x^(3/2)*exp(4*b*x^(1/2)),y(x), singsol=all)
\[ \frac {\sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\, \sqrt {x}\, \operatorname {BesselI}\left (1, \sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\right ) c_{1} b -\operatorname {BesselK}\left (1, -\sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\, b \sqrt {x}-\operatorname {BesselI}\left (0, \sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\right ) c_{1} +\operatorname {BesselK}\left (0, -\sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\right )}{\operatorname {BesselI}\left (1, \sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\, b \sqrt {x}-\operatorname {BesselI}\left (0, \sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\right )} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]+a*(1+2*b*x^(1/2))*Exp[2*b*x^(1/2)]*y[x]==-a^2*b*x^(3/2)*exp(4*b*x^(1/2)),y[x],x,IncludeSingularSolutions -> True]
Not solved