ODE
\[ x (\text {a0}+\text {b0} x) y'''(x)+(\text {a1}+\text {b1} x) y''(x)+x y'(x)+y(x)=f(x) \] ODE Classification
[[_3rd_order, _exact, _linear, _nonhomogeneous]]
Book solution method
TO DO
Mathematica ✗
cpu = 2.38252 (sec), leaf count = 0 , could not solve
DSolve[y[x] + x*Derivative[1][y][x] + (a1 + b1*x)*Derivative[2][y][x] + x*(a0 + b0*x)*Derivative[3][y][x] == f[x], y[x], x]
Maple ✓
cpu = 0.614 (sec), leaf count = 1221
\[ \left \{ y \left ( x \right ) = \left ( -{\it HeunC} \left ( 0,{\frac {2\,{\it a0}-{\it a1}}{{\it a0}}},{\frac {{\it a0}\, \left ( -{\it b1}+2\,{\it b0} \right ) +{\it a1}\,{\it b0}}{{\it a0}\,{\it b0}}},-{\frac {{\it a0}}{{{\it b0}}^{2}}},{\frac { \left ( 4\,{\it b0}-{\it b1} \right ) {{\it a0}}^{2}-{\it a1}\,{\it b1}\,{\it a0}+{{\it a1}}^{2}{\it b0}}{2\,{{\it a0}}^{2}{\it b0}}},-{\frac {{\it b0}\,x}{{\it a0}}} \right ) \left ( \int \!{({\it \_C1}+\int \!f \left ( x \right ) \,{\rm d}x){\it HeunC} \left ( 0,{\frac {-2\,{\it a0}+{\it a1}}{{\it a0}}},{\frac {{\it a0}\, \left ( -{\it b1}+2\,{\it b0} \right ) +{\it a1}\,{\it b0}}{{\it a0}\,{\it b0}}},-{\frac {{\it a0}}{{{\it b0}}^{2}}},{\frac { \left ( 4\,{\it b0}-{\it b1} \right ) {{\it a0}}^{2}-{\it a1}\,{\it b1}\,{\it a0}+{{\it a1}}^{2}{\it b0}}{2\,{{\it a0}}^{2}{\it b0}}},-{\frac {{\it b0}\,x}{{\it a0}}} \right ) \left ( {\it b0}\,x+{\it a0} \right ) ^{{\frac { \left ( -3\,{\it b0}+{\it b1} \right ) {\it a0}-{\it a1}\,{\it b0}}{{\it a0}\,{\it b0}}}}{x}^{{\frac {-2\,{\it a0}+{\it a1}}{{\it a0}}}} \left ( \left ( \left ( -2\,{\it a0}+{\it a1} \right ) {\it HeunC} \left ( 0,{\frac {-2\,{\it a0}+{\it a1}}{{\it a0}}},{\frac {{\it a0}\, \left ( -{\it b1}+2\,{\it b0} \right ) +{\it a1}\,{\it b0}}{{\it a0}\,{\it b0}}},-{\frac {{\it a0}}{{{\it b0}}^{2}}},{\frac { \left ( 4\,{\it b0}-{\it b1} \right ) {{\it a0}}^{2}-{\it a1}\,{\it b1}\,{\it a0}+{{\it a1}}^{2}{\it b0}}{2\,{{\it a0}}^{2}{\it b0}}},-{\frac {{\it b0}\,x}{{\it a0}}} \right ) -x{\it HeunCPrime} \left ( 0,{\frac {-2\,{\it a0}+{\it a1}}{{\it a0}}},{\frac {{\it a0}\, \left ( -{\it b1}+2\,{\it b0} \right ) +{\it a1}\,{\it b0}}{{\it a0}\,{\it b0}}},-{\frac {{\it a0}}{{{\it b0}}^{2}}},{\frac { \left ( 4\,{\it b0}-{\it b1} \right ) {{\it a0}}^{2}-{\it a1}\,{\it b1}\,{\it a0}+{{\it a1}}^{2}{\it b0}}{2\,{{\it a0}}^{2}{\it b0}}},-{\frac {{\it b0}\,x}{{\it a0}}} \right ) {\it b0} \right ) {\it HeunC} \left ( 0,{\frac {2\,{\it a0}-{\it a1}}{{\it a0}}},{\frac {{\it a0}\, \left ( -{\it b1}+2\,{\it b0} \right ) +{\it a1}\,{\it b0}}{{\it a0}\,{\it b0}}},-{\frac {{\it a0}}{{{\it b0}}^{2}}},{\frac { \left ( 4\,{\it b0}-{\it b1} \right ) {{\it a0}}^{2}-{\it a1}\,{\it b1}\,{\it a0}+{{\it a1}}^{2}{\it b0}}{2\,{{\it a0}}^{2}{\it b0}}},-{\frac {{\it b0}\,x}{{\it a0}}} \right ) +{\it HeunCPrime} \left ( 0,{\frac {2\,{\it a0}-{\it a1}}{{\it a0}}},{\frac {{\it a0}\, \left ( -{\it b1}+2\,{\it b0} \right ) +{\it a1}\,{\it b0}}{{\it a0}\,{\it b0}}},-{\frac {{\it a0}}{{{\it b0}}^{2}}},{\frac { \left ( 4\,{\it b0}-{\it b1} \right ) {{\it a0}}^{2}-{\it a1}\,{\it b1}\,{\it a0}+{{\it a1}}^{2}{\it b0}}{2\,{{\it a0}}^{2}{\it b0}}},-{\frac {{\it b0}\,x}{{\it a0}}} \right ) {\it HeunC} \left ( 0,{\frac {-2\,{\it a0}+{\it a1}}{{\it a0}}},{\frac {{\it a0}\, \left ( -{\it b1}+2\,{\it b0} \right ) +{\it a1}\,{\it b0}}{{\it a0}\,{\it b0}}},-{\frac {{\it a0}}{{{\it b0}}^{2}}},{\frac { \left ( 4\,{\it b0}-{\it b1} \right ) {{\it a0}}^{2}-{\it a1}\,{\it b1}\,{\it a0}+{{\it a1}}^{2}{\it b0}}{2\,{{\it a0}}^{2}{\it b0}}},-{\frac {{\it b0}\,x}{{\it a0}}} \right ) x{\it b0} \right ) ^{-1}}\,{\rm d}x{\it a0}-{\it \_C2} \right ) {x}^{{\frac {2\,{\it a0}-{\it a1}}{{\it a0}}}}+{\it HeunC} \left ( 0,{\frac {-2\,{\it a0}+{\it a1}}{{\it a0}}},{\frac {{\it a0}\, \left ( -{\it b1}+2\,{\it b0} \right ) +{\it a1}\,{\it b0}}{{\it a0}\,{\it b0}}},-{\frac {{\it a0}}{{{\it b0}}^{2}}},{\frac { \left ( 4\,{\it b0}-{\it b1} \right ) {{\it a0}}^{2}-{\it a1}\,{\it b1}\,{\it a0}+{{\it a1}}^{2}{\it b0}}{2\,{{\it a0}}^{2}{\it b0}}},-{\frac {{\it b0}\,x}{{\it a0}}} \right ) \left ( \int \!{({\it \_C1}+\int \!f \left ( x \right ) \,{\rm d}x){\it HeunC} \left ( 0,{\frac {2\,{\it a0}-{\it a1}}{{\it a0}}},{\frac {{\it a0}\, \left ( -{\it b1}+2\,{\it b0} \right ) +{\it a1}\,{\it b0}}{{\it a0}\,{\it b0}}},-{\frac {{\it a0}}{{{\it b0}}^{2}}},{\frac { \left ( 4\,{\it b0}-{\it b1} \right ) {{\it a0}}^{2}-{\it a1}\,{\it b1}\,{\it a0}+{{\it a1}}^{2}{\it b0}}{2\,{{\it a0}}^{2}{\it b0}}},-{\frac {{\it b0}\,x}{{\it a0}}} \right ) \left ( {\it b0}\,x+{\it a0} \right ) ^{{\frac { \left ( -3\,{\it b0}+{\it b1} \right ) {\it a0}-{\it a1}\,{\it b0}}{{\it a0}\,{\it b0}}}} \left ( \left ( \left ( -2\,{\it a0}+{\it a1} \right ) {\it HeunC} \left ( 0,{\frac {-2\,{\it a0}+{\it a1}}{{\it a0}}},{\frac {{\it a0}\, \left ( -{\it b1}+2\,{\it b0} \right ) +{\it a1}\,{\it b0}}{{\it a0}\,{\it b0}}},-{\frac {{\it a0}}{{{\it b0}}^{2}}},{\frac { \left ( 4\,{\it b0}-{\it b1} \right ) {{\it a0}}^{2}-{\it a1}\,{\it b1}\,{\it a0}+{{\it a1}}^{2}{\it b0}}{2\,{{\it a0}}^{2}{\it b0}}},-{\frac {{\it b0}\,x}{{\it a0}}} \right ) -x{\it HeunCPrime} \left ( 0,{\frac {-2\,{\it a0}+{\it a1}}{{\it a0}}},{\frac {{\it a0}\, \left ( -{\it b1}+2\,{\it b0} \right ) +{\it a1}\,{\it b0}}{{\it a0}\,{\it b0}}},-{\frac {{\it a0}}{{{\it b0}}^{2}}},{\frac { \left ( 4\,{\it b0}-{\it b1} \right ) {{\it a0}}^{2}-{\it a1}\,{\it b1}\,{\it a0}+{{\it a1}}^{2}{\it b0}}{2\,{{\it a0}}^{2}{\it b0}}},-{\frac {{\it b0}\,x}{{\it a0}}} \right ) {\it b0} \right ) {\it HeunC} \left ( 0,{\frac {2\,{\it a0}-{\it a1}}{{\it a0}}},{\frac {{\it a0}\, \left ( -{\it b1}+2\,{\it b0} \right ) +{\it a1}\,{\it b0}}{{\it a0}\,{\it b0}}},-{\frac {{\it a0}}{{{\it b0}}^{2}}},{\frac { \left ( 4\,{\it b0}-{\it b1} \right ) {{\it a0}}^{2}-{\it a1}\,{\it b1}\,{\it a0}+{{\it a1}}^{2}{\it b0}}{2\,{{\it a0}}^{2}{\it b0}}},-{\frac {{\it b0}\,x}{{\it a0}}} \right ) +{\it HeunCPrime} \left ( 0,{\frac {2\,{\it a0}-{\it a1}}{{\it a0}}},{\frac {{\it a0}\, \left ( -{\it b1}+2\,{\it b0} \right ) +{\it a1}\,{\it b0}}{{\it a0}\,{\it b0}}},-{\frac {{\it a0}}{{{\it b0}}^{2}}},{\frac { \left ( 4\,{\it b0}-{\it b1} \right ) {{\it a0}}^{2}-{\it a1}\,{\it b1}\,{\it a0}+{{\it a1}}^{2}{\it b0}}{2\,{{\it a0}}^{2}{\it b0}}},-{\frac {{\it b0}\,x}{{\it a0}}} \right ) {\it HeunC} \left ( 0,{\frac {-2\,{\it a0}+{\it a1}}{{\it a0}}},{\frac {{\it a0}\, \left ( -{\it b1}+2\,{\it b0} \right ) +{\it a1}\,{\it b0}}{{\it a0}\,{\it b0}}},-{\frac {{\it a0}}{{{\it b0}}^{2}}},{\frac { \left ( 4\,{\it b0}-{\it b1} \right ) {{\it a0}}^{2}-{\it a1}\,{\it b1}\,{\it a0}+{{\it a1}}^{2}{\it b0}}{2\,{{\it a0}}^{2}{\it b0}}},-{\frac {{\it b0}\,x}{{\it a0}}} \right ) x{\it b0} \right ) ^{-1}}\,{\rm d}x{\it a0}+{\it \_C3} \right ) \right ) \left ( {\it b0}\,x+{\it a0} \right ) ^{{\frac {{\it a0}\, \left ( -{\it b1}+2\,{\it b0} \right ) +{\it a1}\,{\it b0}}{{\it a0}\,{\it b0}}}} \right \} \] Mathematica raw input
DSolve[y[x] + x*y'[x] + (a1 + b1*x)*y''[x] + x*(a0 + b0*x)*y'''[x] == f[x],y[x],x]
Mathematica raw output
DSolve[y[x] + x*Derivative[1][y][x] + (a1 + b1*x)*Derivative[2][y][x] + x*(a0 + b0*x)*Derivative[3][y][x] == f[x], y[x], x]
Maple raw input
dsolve(x*(b0*x+a0)*diff(diff(diff(y(x),x),x),x)+(b1*x+a1)*diff(diff(y(x),x),x)+x*diff(y(x),x)+y(x) = f(x), y(x),'implicit')
Maple raw output
y(x) = (-HeunC(0,(2*a0-a1)/a0,(a0*(-b1+2*b0)+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)*(Int(HeunC(0,(-2*a0+a1)/a0,(a0*(-b1+2*b0)+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)*(b0*x+a0)^(((-3*b0+b1)*a0-a1*b0)/a0/b0)*(_C1+Int(f(x),x))*x^((-2*a0+a1)/a0)/(((-2*a0+a1)*HeunC(0,(-2*a0+a1)/a0,(a0*(-b1+2*b0)+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)-x*HeunCPrime(0,(-2*a0+a1)/a0,(a0*(-b1+2*b0)+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)*b0)*HeunC(0,(2*a0-a1)/a0,(a0*(-b1+2*b0)+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)+HeunCPrime(0,(2*a0-a1)/a0,(a0*(-b1+2*b0)+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)*HeunC(0,(-2*a0+a1)/a0,(a0*(-b1+2*b0)+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)*x*b0),x)*a0-_C2)*x^((2*a0-a1)/a0)+HeunC(0,(-2*a0+a1)/a0,(a0*(-b1+2*b0)+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)*(Int(HeunC(0,(2*a0-a1)/a0,(a0*(-b1+2*b0)+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)*(b0*x+a0)^(((-3*b0+b1)*a0-a1*b0)/a0/b0)*(_C1+Int(f(x),x))/(((-2*a0+a1)*HeunC(0,(-2*a0+a1)/a0,(a0*(-b1+2*b0)+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)-x*HeunCPrime(0,(-2*a0+a1)/a0,(a0*(-b1+2*b0)+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)*b0)*HeunC(0,(2*a0-a1)/a0,(a0*(-b1+2*b0)+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)+HeunCPrime(0,(2*a0-a1)/a0,(a0*(-b1+2*b0)+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)*HeunC(0,(-2*a0+a1)/a0,(a0*(-b1+2*b0)+a1*b0)/a0/b0,-1/b0^2*a0,1/2*((4*b0-b1)*a0^2-a1*b1*a0+a1^2*b0)/a0^2/b0,-b0/a0*x)*x*b0),x)*a0+_C3))*(b0*x+a0)^((a0*(-b1+2*b0)+a1*b0)/a0/b0)