4.26.48 \((a+b x) y'(x)+c y(x)+y''(x)=0\)

ODE
\[ (a+b x) y'(x)+c y(x)+y''(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.0277966 (sec), leaf count = 81

\[\left \{\left \{y(x)\to e^{-\frac {1}{2} x (2 a+b x)} \left (c_1 H_{\frac {c}{b}-1}\left (\frac {a+b x}{\sqrt {2} \sqrt {b}}\right )+c_2 \, _1F_1\left (\frac {b-c}{2 b};\frac {1}{2};\frac {(a+b x)^2}{2 b}\right )\right )\right \}\right \}\]

Maple ✓
cpu = 0.044 (sec), leaf count = 49

\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{{\sl M}\left ({\frac {c}{2\,b}},\,{\frac {1}{2}},\,-{\frac { \left ( bx+a \right ) ^{2}}{2\,b}}\right )}+{\it \_C2}\,{{\sl U}\left ({\frac {c}{2\,b}},\,{\frac {1}{2}},\,-{\frac { \left ( bx+a \right ) ^{2}}{2\,b}}\right )} \right \} \] Mathematica raw input

DSolve[c*y[x] + (a + b*x)*y'[x] + y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (C[1]*HermiteH[-1 + c/b, (a + b*x)/(Sqrt[2]*Sqrt[b])] + C[2]*Hypergeom
etric1F1[(b - c)/(2*b), 1/2, (a + b*x)^2/(2*b)])/E^((x*(2*a + b*x))/2)}}

Maple raw input

dsolve(diff(diff(y(x),x),x)+(b*x+a)*diff(y(x),x)+c*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = _C1*KummerM(1/2*c/b,1/2,-1/2/b*(b*x+a)^2)+_C2*KummerU(1/2*c/b,1/2,-1/2/b*
(b*x+a)^2)