ODE
\[ y(x) \left (a+b x+c x^2\right )+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.025916 (sec), leaf count = 110
\[\left \{\left \{y(x)\to c_2 D_{\frac {-i b^2-4 c^{3/2}+4 i a c}{8 c^{3/2}}}\left (-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) (b+2 c x)}{c^{3/4}}\right )+c_1 D_{\frac {i \left (b^2+4 i c^{3/2}-4 a c\right )}{8 c^{3/2}}}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) (b+2 c x)}{c^{3/4}}\right )\right \}\right \}\]
Maple ✓
cpu = 0.108 (sec), leaf count = 109
\[ \left \{ y \left ( x \right ) = \left ( {\it \_C2}\, \left ( 2\,cx+b \right ) {\mbox {$_1$F$_1$}({\frac {1}{16} \left ( 12\,{c}^{3/2}+4\,iac-i{b}^{2} \right ) {c}^{-{\frac {3}{2}}}};\,{\frac {3}{2}};\,{{\frac {i}{4}} \left ( 2\,cx+b \right ) ^{2}{c}^{-{\frac {3}{2}}}})}+{\it \_C1}\,{\mbox {$_1$F$_1$}({\frac {1}{16} \left ( 4\,{c}^{3/2}+4\,iac-i{b}^{2} \right ) {c}^{-{\frac {3}{2}}}};\,{\frac {1}{2}};\,{{\frac {i}{4}} \left ( 2\,cx+b \right ) ^{2}{c}^{-{\frac {3}{2}}}})} \right ) {{\rm e}^{{-{\frac {i}{2}}x \left ( cx+b \right ) {\frac {1}{\sqrt {c}}}}}} \right \} \] Mathematica raw input
DSolve[(a + b*x + c*x^2)*y[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[2]*ParabolicCylinderD[((-I)*b^2 + (4*I)*a*c - 4*c^(3/2))/(8*c^(3/2)), ((-1/2 + I/2)*(b + 2*c*x))/c^(3/4)] + C[1]*ParabolicCylinderD[((I/8)*(b^2 - 4*a*c + (4*I)*c^(3/2)))/c^(3/2), ((1/2 + I/2)*(b + 2*c*x))/c^(3/4)]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+(c*x^2+b*x+a)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (_C2*(2*c*x+b)*hypergeom([1/16*(12*c^(3/2)+4*I*a*c-I*b^2)/c^(3/2)],[3/2],1/4*I*(2*c*x+b)^2/c^(3/2))+_C1*hypergeom([1/16*(4*c^(3/2)+4*I*a*c-I*b^2)/c^(3/2)],[1/2],1/4*I*(2*c*x+b)^2/c^(3/2)))*exp(-1/2*I*x*(c*x+b)/c^(1/2))