Internal problem ID [9388]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1055.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x +\operatorname {c1} \right ) y=0} \]
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 254
dsolve(diff(diff(y(x),x),x)+(a*x+b)*diff(y(x),x)+(a1*x^2+b1*x+c1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (c_{2} \left (a^{2} x +a b -4 \operatorname {a1} x -2 \operatorname {b1} \right ) \operatorname {hypergeom}\left (\left [\frac {3 \left (a^{2}-4 \operatorname {a1} \right )^{\frac {3}{2}}+a^{3}-2 a^{2} \operatorname {c1} +2 \left (\operatorname {b1} b -2 \operatorname {a1} \right ) a +2 \left (-b^{2}+4 \operatorname {c1} \right ) \operatorname {a1} -2 \operatorname {b1}^{2}}{4 \left (a^{2}-4 \operatorname {a1} \right )^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {\left (a^{2} x +a b -4 \operatorname {a1} x -2 \operatorname {b1} \right )^{2}}{2 \left (a^{2}-4 \operatorname {a1} \right )^{\frac {3}{2}}}\right )+\operatorname {hypergeom}\left (\left [\frac {\left (a^{2}-4 \operatorname {a1} \right )^{\frac {3}{2}}+a^{3}-2 a^{2} \operatorname {c1} +\left (2 \operatorname {b1} b -4 \operatorname {a1} \right ) a +\left (-2 b^{2}+8 \operatorname {c1} \right ) \operatorname {a1} -2 \operatorname {b1}^{2}}{4 \left (a^{2}-4 \operatorname {a1} \right )^{\frac {3}{2}}}\right ], \left [\frac {1}{2}\right ], \frac {\left (a^{2} x +a b -4 \operatorname {a1} x -2 \operatorname {b1} \right )^{2}}{2 \left (a^{2}-4 \operatorname {a1} \right )^{\frac {3}{2}}}\right ) c_{1} \right ) {\mathrm e}^{-\frac {x \left (\left (a x +2 b \right ) \sqrt {a^{2}-4 \operatorname {a1}}+x \left (a^{2}-4 \operatorname {a1} \right )+2 a b -4 \operatorname {b1} \right )}{4 \sqrt {a^{2}-4 \operatorname {a1}}}} \]
✓ Solution by Mathematica
Time used: 0.278 (sec). Leaf size: 305
DSolve[(c1 + b1*x + a1*x^2)*y[x] + (b + a*x)*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \exp \left (-\frac {x \left (a \left (x \sqrt {a^2-4 \text {a1}}+2 b\right )+2 b \sqrt {a^2-4 \text {a1}}+a^2 x-4 (\text {a1} x+\text {b1})\right )}{4 \sqrt {a^2-4 \text {a1}}}\right ) \left (c_1 \operatorname {HermiteH}\left (\frac {-a^3-\left (\sqrt {a^2-4 \text {a1}}-2 \text {c1}\right ) a^2+(4 \text {a1}-2 b \text {b1}) a+2 \left (\text {b1}^2+\text {a1} \left (b^2-4 \text {c1}+2 \sqrt {a^2-4 \text {a1}}\right )\right )}{2 \left (a^2-4 \text {a1}\right )^{3/2}},\frac {x a^2+b a-2 (\text {b1}+2 \text {a1} x)}{\sqrt {2} \left (a^2-4 \text {a1}\right )^{3/4}}\right )+c_2 \operatorname {Hypergeometric1F1}\left (\frac {a^3+\left (\sqrt {a^2-4 \text {a1}}-2 \text {c1}\right ) a^2+(2 b \text {b1}-4 \text {a1}) a-2 \left (\text {b1}^2+\text {a1} \left (b^2-4 \text {c1}+2 \sqrt {a^2-4 \text {a1}}\right )\right )}{4 \left (a^2-4 \text {a1}\right )^{3/2}},\frac {1}{2},\frac {\left (x a^2+b a-2 (\text {b1}+2 \text {a1} x)\right )^2}{2 \left (a^2-4 \text {a1}\right )^{3/2}}\right )\right ) \]