ODE
\[ (\text {a0}+\text {b0} x) y'(x)+y(x) (\text {a1}+\text {b1} x)+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.19151 (sec), leaf count = 132
\[\left \{\left \{y(x)\to e^{-\text {a0} x+\frac {\text {b1} x}{\text {b0}}-\frac {\text {b0} x^2}{2}} \left (c_2 \, _1F_1\left (\frac {\text {b0}^3-\text {a1} \text {b0}^2+\text {a0} \text {b1} \text {b0}-\text {b1}^2}{2 \text {b0}^3};\frac {1}{2};\frac {\left (x \text {b0}^2+\text {a0} \text {b0}-2 \text {b1}\right )^2}{2 \text {b0}^3}\right )+c_1 H_{\frac {-\text {b0}^3+\text {a1} \text {b0}^2-\text {a0} \text {b1} \text {b0}+\text {b1}^2}{\text {b0}^3}}\left (\frac {x \text {b0}^2+\text {a0} \text {b0}-2 \text {b1}}{\sqrt {2} \text {b0}^{3/2}}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.152 (sec), leaf count = 105
\[\left [y \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{-\frac {\mathit {b1} x}{\mathit {b0}}} \KummerM \left (\frac {-\mathit {a0} \mathit {b0} \mathit {b1} +\mathit {a1} \,\mathit {b0}^{2}+\mathit {b1}^{2}}{2 \mathit {b0}^{3}}, \frac {1}{2}, -\frac {\left (x \,\mathit {b0}^{2}+\mathit {a0} \mathit {b0} -2 \mathit {b1} \right )^{2}}{2 \mathit {b0}^{3}}\right )+\textit {\_C2} \,{\mathrm e}^{-\frac {\mathit {b1} x}{\mathit {b0}}} \KummerU \left (\frac {-\mathit {a0} \mathit {b0} \mathit {b1} +\mathit {a1} \,\mathit {b0}^{2}+\mathit {b1}^{2}}{2 \mathit {b0}^{3}}, \frac {1}{2}, -\frac {\left (x \,\mathit {b0}^{2}+\mathit {a0} \mathit {b0} -2 \mathit {b1} \right )^{2}}{2 \mathit {b0}^{3}}\right )\right ]\] Mathematica raw input
DSolve[(a1 + b1*x)*y[x] + (a0 + b0*x)*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> E^(-(a0*x) + (b1*x)/b0 - (b0*x^2)/2)*(C[1]*HermiteH[(a1*b0^2 - b0^3 - a0*b0*b1 + b1^2)/b0^3, (a0*b0 - 2*b1 + b0^2*x)/(Sqrt[2]*b0^(3/2))] + C[2]*Hypergeometric1F1[(-(a1*b0^2) + b0^3 + a0*b0*b1 - b1^2)/(2*b0^3), 1/2, (a0*b0 - 2*b1 + b0^2*x)^2/(2*b0^3)])}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+(b0*x+a0)*diff(y(x),x)+(b1*x+a1)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*exp(-b1/b0*x)*KummerM(1/2*(-a0*b0*b1+a1*b0^2+b1^2)/b0^3,1/2,-1/2/b0^3*(b0^2*x+a0*b0-2*b1)^2)+_C2*exp(-b1/b0*x)*KummerU(1/2*(-a0*b0*b1+a1*b0^2+b1^2)/b0^3,1/2,-1/2/b0^3*(b0^2*x+a0*b0-2*b1)^2)]