29.7.4 problem 179

Internal problem ID [4779]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 7
Problem number : 179
Date solved : Monday, January 27, 2025 at 09:37:28 AM
CAS classification : [_rational, _Riccati]

\begin{align*} x y^{\prime }+\operatorname {a0} +\operatorname {a1} x +\left (\operatorname {a2} +\operatorname {a3} x y\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.029 (sec). Leaf size: 403

dsolve(x*diff(y(x),x)+a0+a1*x+(a2+a3*x*y(x))*y(x) = 0,y(x), singsol=all)
 

\[ y \left (x \right ) = -\frac {4 \operatorname {a1} \left (\operatorname {a1}^{3} \operatorname {a3} \left (\operatorname {a3} \operatorname {a0} -\operatorname {a2} \sqrt {-\operatorname {a1} \operatorname {a3}}\right ) \operatorname {KummerM}\left (\frac {\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \left (\operatorname {a2} +2\right )}{2 \operatorname {a1}}, \operatorname {a2} +1, 2 x \sqrt {-\operatorname {a1} \operatorname {a3}}\right )-\frac {c_{1} \left (\operatorname {a0}^{2} \operatorname {a3} +\operatorname {a1} \,\operatorname {a2}^{2}\right ) \operatorname {KummerU}\left (\frac {\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \left (\operatorname {a2} +2\right )}{2 \operatorname {a1}}, \operatorname {a2} +1, 2 x \sqrt {-\operatorname {a1} \operatorname {a3}}\right )}{4}+\operatorname {a1}^{3} \operatorname {a3} \left (\operatorname {a2} \sqrt {-\operatorname {a1} \operatorname {a3}}+\operatorname {a3} \operatorname {a0} \right ) \operatorname {KummerM}\left (\frac {\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \operatorname {a2}}{2 \operatorname {a1}}, \operatorname {a2} +1, 2 x \sqrt {-\operatorname {a1} \operatorname {a3}}\right )+\frac {\operatorname {KummerU}\left (\frac {\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \operatorname {a2}}{2 \operatorname {a1}}, \operatorname {a2} +1, 2 x \sqrt {-\operatorname {a1} \operatorname {a3}}\right ) c_{1} \left (\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} -\operatorname {a1} \operatorname {a2} \right )}{2}\right )}{4 \operatorname {a1}^{3} \operatorname {a3}^{2} \left (\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \operatorname {a2} \right ) \operatorname {KummerM}\left (\frac {\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \left (\operatorname {a2} +2\right )}{2 \operatorname {a1}}, \operatorname {a2} +1, 2 x \sqrt {-\operatorname {a1} \operatorname {a3}}\right )-c_{1} \sqrt {-\operatorname {a1} \operatorname {a3}}\, \left (\operatorname {a0}^{2} \operatorname {a3} +\operatorname {a1} \,\operatorname {a2}^{2}\right ) \operatorname {KummerU}\left (\frac {\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \left (\operatorname {a2} +2\right )}{2 \operatorname {a1}}, \operatorname {a2} +1, 2 x \sqrt {-\operatorname {a1} \operatorname {a3}}\right )-4 \operatorname {a1} \left (\operatorname {a1}^{2} \operatorname {a3}^{2} \left (\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} -\operatorname {a1} \operatorname {a2} \right ) \operatorname {KummerM}\left (\frac {\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \operatorname {a2}}{2 \operatorname {a1}}, \operatorname {a2} +1, 2 x \sqrt {-\operatorname {a1} \operatorname {a3}}\right )-\frac {\operatorname {KummerU}\left (\frac {\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \operatorname {a2}}{2 \operatorname {a1}}, \operatorname {a2} +1, 2 x \sqrt {-\operatorname {a1} \operatorname {a3}}\right ) c_{1} \left (\operatorname {a2} \sqrt {-\operatorname {a1} \operatorname {a3}}+\operatorname {a3} \operatorname {a0} \right )}{2}\right )} \]

✓ Solution by Mathematica

Time used: 0.445 (sec). Leaf size: 421

DSolve[x D[y[x],x]+a0+a1 x+(a2+a3 x y[x])y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {i \left (\sqrt {\text {a1}} c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}\right ),\text {a2},2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )+c_1 \left (\sqrt {\text {a1}} \text {a2}+i \text {a0} \sqrt {\text {a3}}\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}+2\right ),\text {a2}+1,2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )+\sqrt {\text {a1}} \left (2 L_{-\frac {i \sqrt {\text {a3}} \text {a0}}{2 \sqrt {\text {a1}}}-\frac {\text {a2}}{2}-1}^{\text {a2}}\left (2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )+L_{-\frac {i \sqrt {\text {a3}} \text {a0}}{2 \sqrt {\text {a1}}}-\frac {\text {a2}}{2}}^{\text {a2}-1}\left (2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )\right )\right )}{\sqrt {\text {a3}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}\right ),\text {a2},2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )+L_{-\frac {i \sqrt {\text {a3}} \text {a0}}{2 \sqrt {\text {a1}}}-\frac {\text {a2}}{2}}^{\text {a2}-1}\left (2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )\right )} \\ y(x)\to \frac {\frac {\left (\text {a0} \sqrt {\text {a3}}-i \sqrt {\text {a1}} \text {a2}\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}+2\right ),\text {a2}+1,2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )}{\operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}\right ),\text {a2},2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )}-i \sqrt {\text {a1}}}{\sqrt {\text {a3}}} \\ \end{align*}