ODE
\[ (\text {a0}+\text {b0} x) y'(x)+y(x) \left (\text {a1}+\text {b1} x+\text {c1} x^2\right )+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.141929 (sec), leaf count = 303
\[\left \{\left \{y(x)\to \exp \left (-\frac {x \left (2 \text {a0} \left (\sqrt {\text {b0}^2-4 \text {c1}}+\text {b0}\right )+x \left (\text {b0} \sqrt {\text {b0}^2-4 \text {c1}}+\text {b0}^2-4 \text {c1}\right )-4 \text {b1}\right )}{4 \sqrt {\text {b0}^2-4 \text {c1}}}\right ) \left (c_1 H_{\frac {2 \text {a0}^2 \text {c1}-2 \text {a0} \text {b0} \text {b1}+2 \text {a1} \left (\text {b0}^2-4 \text {c1}\right )-\text {b0}^3-\text {b0}^2 \sqrt {\text {b0}^2-4 \text {c1}}+4 \text {c1} \sqrt {\text {b0}^2-4 \text {c1}}+4 \text {b0} \text {c1}+2 \text {b1}^2}{2 \left (\text {b0}^2-4 \text {c1}\right )^{3/2}}}\left (\frac {\text {a0} \text {b0}+\text {b0}^2 x-2 \text {b1}-4 \text {c1} x}{\sqrt {2} \left (\text {b0}^2-4 \text {c1}\right )^{3/4}}\right )+c_2 \, _1F_1\left (\frac {\text {b0}^3-2 \text {a1} \text {b0}^2+\sqrt {\text {b0}^2-4 \text {c1}} \text {b0}^2+2 \text {a0} \text {b1} \text {b0}-4 \text {c1} \text {b0}-2 \text {b1}^2-2 \text {a0}^2 \text {c1}+8 \text {a1} \text {c1}-4 \sqrt {\text {b0}^2-4 \text {c1}} \text {c1}}{4 \left (\text {b0}^2-4 \text {c1}\right )^{3/2}};\frac {1}{2};\frac {\left (x \text {b0}^2+\text {a0} \text {b0}-2 \text {b1}-4 \text {c1} x\right )^2}{2 \left (\text {b0}^2-4 \text {c1}\right )^{3/2}}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.319 (sec), leaf count = 258
\[ \left \{ y \left ( x \right ) = \left ( {\it \_C2}\, \left ( x{{\it b0}}^{2}+{\it a0}\,{\it b0}-4\,{\it c1}\,x-2\,{\it b1} \right ) {\mbox {$_1$F$_1$}(-{\frac {1}{2} \left ( -{\frac {3}{2} \left ( {{\it b0}}^{2}-4\,{\it c1} \right ) ^{{\frac {3}{2}}}}-{\frac {{{\it b0}}^{3}}{2}}+{\it a1}\,{{\it b0}}^{2}+ \left ( -{\it a0}\,{\it b1}+2\,{\it c1} \right ) {\it b0}+ \left ( {{\it a0}}^{2}-4\,{\it a1} \right ) {\it c1}+{{\it b1}}^{2} \right ) \left ( {{\it b0}}^{2}-4\,{\it c1} \right ) ^{-{\frac {3}{2}}}};\,{\frac {3}{2}};\,{\frac { \left ( x{{\it b0}}^{2}+{\it a0}\,{\it b0}-4\,{\it c1}\,x-2\,{\it b1} \right ) ^{2}}{2} \left ( {{\it b0}}^{2}-4\,{\it c1} \right ) ^{-{\frac {3}{2}}}})}+{\it \_C1}\,{\mbox {$_1$F$_1$}(-{\frac {1}{2} \left ( -{\frac {1}{2} \left ( {{\it b0}}^{2}-4\,{\it c1} \right ) ^{{\frac {3}{2}}}}-{\frac {{{\it b0}}^{3}}{2}}+{\it a1}\,{{\it b0}}^{2}+ \left ( -{\it a0}\,{\it b1}+2\,{\it c1} \right ) {\it b0}+ \left ( {{\it a0}}^{2}-4\,{\it a1} \right ) {\it c1}+{{\it b1}}^{2} \right ) \left ( {{\it b0}}^{2}-4\,{\it c1} \right ) ^{-{\frac {3}{2}}}};\,{\frac {1}{2}};\,{\frac { \left ( x{{\it b0}}^{2}+{\it a0}\,{\it b0}-4\,{\it c1}\,x-2\,{\it b1} \right ) ^{2}}{2} \left ( {{\it b0}}^{2}-4\,{\it c1} \right ) ^{-{\frac {3}{2}}}})} \right ) {{\rm e}^{-{\frac {x}{4} \left ( \left ( {\it b0}\,x+2\,{\it a0} \right ) \left ( {{\it b0}}^{2}-4\,{\it c1} \right ) ^{{\frac {3}{2}}}+ \left ( {{\it b0}}^{2}-4\,{\it c1} \right ) \left ( x{{\it b0}}^{2}+2\,{\it a0}\,{\it b0}-4\,{\it c1}\,x-4\,{\it b1} \right ) \right ) \left ( {{\it b0}}^{2}-4\,{\it c1} \right ) ^{-{\frac {3}{2}}}}}} \right \} \] Mathematica raw input
DSolve[(a1 + b1*x + c1*x^2)*y[x] + (a0 + b0*x)*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (C[1]*HermiteH[(-b0^3 - 2*a0*b0*b1 + 2*b1^2 - b0^2*Sqrt[b0^2 - 4*c1] + 2*a1*(b0^2 - 4*c1) + 2*a0^2*c1 + 4*b0*c1 + 4*Sqrt[b0^2 - 4*c1]*c1)/(2*(b0^2 - 4*c1)^(3/2)), (a0*b0 - 2*b1 + b0^2*x - 4*c1*x)/(Sqrt[2]*(b0^2 - 4*c1)^(3/4))] + C[2]*Hypergeometric1F1[(-2*a1*b0^2 + b0^3 + 2*a0*b0*b1 - 2*b1^2 + b0^2*Sqrt[b0^2 - 4*c1] - 2*a0^2*c1 + 8*a1*c1 - 4*b0*c1 - 4*Sqrt[b0^2 - 4*c1]*c1)/(4*(b0^2 - 4*c1)^(3/2)), 1/2, (a0*b0 - 2*b1 + b0^2*x - 4*c1*x)^2/(2*(b0^2 - 4*c1)^(3/2))])/E^((x*(-4*b1 + 2*a0*(b0 + Sqrt[b0^2 - 4*c1]) + (b0^2 + b0*Sqrt[b0^2 - 4*c1] - 4*c1)*x))/(4*Sqrt[b0^2 - 4*c1]))}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+(b0*x+a0)*diff(y(x),x)+(c1*x^2+b1*x+a1)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (_C2*(b0^2*x+a0*b0-4*c1*x-2*b1)*hypergeom([-1/2/(b0^2-4*c1)^(3/2)*(-3/2*(b0^2-4*c1)^(3/2)-1/2*b0^3+a1*b0^2+(-a0*b1+2*c1)*b0+(a0^2-4*a1)*c1+b1^2)],[3/2],1/2*(b0^2*x+a0*b0-4*c1*x-2*b1)^2/(b0^2-4*c1)^(3/2))+_C1*hypergeom([-1/2/(b0^2-4*c1)^(3/2)*(-1/2*(b0^2-4*c1)^(3/2)-1/2*b0^3+a1*b0^2+(-a0*b1+2*c1)*b0+(a0^2-4*a1)*c1+b1^2)],[1/2],1/2*(b0^2*x+a0*b0-4*c1*x-2*b1)^2/(b0^2-4*c1)^(3/2)))*exp(-1/4/(b0^2-4*c1)^(3/2)*x*((b0*x+2*a0)*(b0^2-4*c1)^(3/2)+(b0^2-4*c1)*(b0^2*x+2*a0*b0-4*c1*x-4*b1)))