y′′(x)( a0 + b0x + c0x2) + (a1 + b1x)y′(x) + 2a2y(x) = 0

4.34.6 $y^{″} (x) (a0 + b0 x + c0 x^{2}) + (a1 + b1 x) y^{'} (x) + 2 a2 y (x) = 0$

ODE
$y^{″} (x) (a0 + b0 x + c0 x^{2}) + (a1 + b1 x) y^{'} (x) + 2 a2 y (x) = 0$ ODE Classiﬁcation

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✗
cpu = 16.2114 (sec), leaf count = 0 , DiﬀerentialRoot result

$\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{2 \text {a2} \unicode {f818}(\unicode {f817})+(\text {a1}+\unicode {f817} \text {b1}) \unicode {f818}'(\unicode {f817})+\left (\text {c0} \unicode {f817}^2+\text {b0} \unicode {f817}+\text {a0}\right ) \unicode {f818}''(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2\right \}\right )(x)\right \}\right \}$

Maple ✓
cpu = 0.245 (sec), leaf count = 507

${y (x) =_C 1_{2} F_{1} (\frac{1}{2 c 0} (- c 0 + b 1 + \sqrt{{c 0}^{2} + (- 8 a 2 - 2 b 1) c 0 + {b 1}^{2}}), - \frac{1}{2 c 0} (c 0 - b 1 + \sqrt{{c 0}^{2} + (- 8 a 2 - 2 b 1) c 0 + {b 1}^{2}}); \frac{1}{2 {c 0}^{2}} (b 1 \sqrt{\frac{- 4 a 0 c 0 + {b 0}^{2}}{{c 0}^{2}}} c 0 - 2 c 0 a 1 + b 1 b 0) \frac{1}{\sqrt{\frac{- 4 a 0 c 0 + {b 0}^{2}}{{c 0}^{2}}}}; \frac{1}{8 a 0 c 0 - 2 {b 0}^{2}} (- 2 \sqrt{\frac{- 4 a 0 c 0 + {b 0}^{2}}{{c 0}^{2}}} x {c 0}^{2} - \sqrt{\frac{- 4 a 0 c 0 + {b 0}^{2}}{{c 0}^{2}}} b 0 c 0 + 4 a 0 c 0 - {b 0}^{2})) +_C 2 {(2 \sqrt{\frac{- 4 a 0 c 0 + {b 0}^{2}}{{c 0}^{2}}} x {c 0}^{2} + \sqrt{\frac{- 4 a 0 c 0 + {b 0}^{2}}{{c 0}^{2}}} b 0 c 0 - 4 a 0 c 0 + {b 0}^{2})}^{\frac{1}{{c 0}^{2}} (- \frac{c 0 (b 1 - 2 c 0)}{2} \sqrt{\frac{- 4 a 0 c 0 + {b 0}^{2}}{{c 0}^{2}}} + c 0 a 1 - \frac{b 1 b 0}{2}) \frac{1}{\sqrt{\frac{- 4 a 0 c 0 + {b 0}^{2}}{{c 0}^{2}}}}}_{2} F_{1} (\frac{1}{{c 0}^{2}} (\frac{c 0}{2} (c 0 - \sqrt{{c 0}^{2} + (- 8 a 2 - 2 b 1) c 0 + {b 1}^{2}}) \sqrt{\frac{- 4 a 0 c 0 + {b 0}^{2}}{{c 0}^{2}}} + c 0 a 1 - \frac{b 1 b 0}{2}) \frac{1}{\sqrt{\frac{- 4 a 0 c 0 + {b 0}^{2}}{{c 0}^{2}}}}, \frac{1}{{c 0}^{2}} (\frac{c 0}{2} (c 0 + \sqrt{{c 0}^{2} + (- 8 a 2 - 2 b 1) c 0 + {b 1}^{2}}) \sqrt{\frac{- 4 a 0 c 0 + {b 0}^{2}}{{c 0}^{2}}} + c 0 a 1 - \frac{b 1 b 0}{2}) \frac{1}{\sqrt{\frac{- 4 a 0 c 0 + {b 0}^{2}}{{c 0}^{2}}}}; \frac{1}{{c 0}^{2}} (- \frac{c 0 (b 1 - 4 c 0)}{2} \sqrt{\frac{- 4 a 0 c 0 + {b 0}^{2}}{{c 0}^{2}}} + c 0 a 1 - \frac{b 1 b 0}{2}) \frac{1}{\sqrt{\frac{- 4 a 0 c 0 + {b 0}^{2}}{{c 0}^{2}}}}; \frac{1}{8 a 0 c 0 - 2 {b 0}^{2}} (- 2 \sqrt{\frac{- 4 a 0 c 0 + {b 0}^{2}}{{c 0}^{2}}} x {c 0}^{2} - \sqrt{\frac{- 4 a 0 c 0 + {b 0}^{2}}{{c 0}^{2}}} b 0 c 0 + 4 a 0 c 0 - {b 0}^{2}))}$ Mathematica raw input

DSolve[2*a2*y[x] + (a1 + b1*x)*y'[x] + (a0 + b0*x + c0*x^2)*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {2*a2*\[FormalY][\
[FormalX]] + (a1 + \[FormalX]*b1)*Derivative[1][\[FormalY]][\[FormalX]] + (a0 +
\[FormalX]*b0 + \[FormalX]^2*c0)*Derivative[2][\[FormalY]][\[FormalX]] == 0, \[F
ormalY][0] == C[1], Derivative[1][\[FormalY]][0] == C[2]}]][x]}}

Maple raw input

dsolve((c0*x^2+b0*x+a0)*diff(diff(y(x),x),x)+(b1*x+a1)*diff(y(x),x)+2*a2*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = _C1*hypergeom([1/2/c0*(-c0+b1+(c0^2+(-8*a2-2*b1)*c0+b1^2)^(1/2)), -1/2/c0
*(c0-b1+(c0^2+(-8*a2-2*b1)*c0+b1^2)^(1/2))],[1/2*(b1*((-4*a0*c0+b0^2)/c0^2)^(1/2
)*c0-2*c0*a1+b1*b0)/c0^2/((-4*a0*c0+b0^2)/c0^2)^(1/2)],(-2*((-4*a0*c0+b0^2)/c0^2
)^(1/2)*x*c0^2-((-4*a0*c0+b0^2)/c0^2)^(1/2)*b0*c0+4*a0*c0-b0^2)/(8*a0*c0-2*b0^2)
)+_C2*(2*((-4*a0*c0+b0^2)/c0^2)^(1/2)*x*c0^2+((-4*a0*c0+b0^2)/c0^2)^(1/2)*b0*c0-
4*a0*c0+b0^2)^(1/((-4*a0*c0+b0^2)/c0^2)^(1/2)*(-1/2*c0*(b1-2*c0)*((-4*a0*c0+b0^2
)/c0^2)^(1/2)+c0*a1-1/2*b1*b0)/c0^2)*hypergeom([(1/2*c0*(c0-(c0^2+(-8*a2-2*b1)*c
0+b1^2)^(1/2))*((-4*a0*c0+b0^2)/c0^2)^(1/2)+c0*a1-1/2*b1*b0)/((-4*a0*c0+b0^2)/c0
^2)^(1/2)/c0^2, (1/2*c0*(c0+(c0^2+(-8*a2-2*b1)*c0+b1^2)^(1/2))*((-4*a0*c0+b0^2)/
c0^2)^(1/2)+c0*a1-1/2*b1*b0)/((-4*a0*c0+b0^2)/c0^2)^(1/2)/c0^2],[1/((-4*a0*c0+b0
^2)/c0^2)^(1/2)*(-1/2*c0*(b1-4*c0)*((-4*a0*c0+b0^2)/c0^2)^(1/2)+c0*a1-1/2*b1*b0)
/c0^2],(-2*((-4*a0*c0+b0^2)/c0^2)^(1/2)*x*c0^2-((-4*a0*c0+b0^2)/c0^2)^(1/2)*b0*c
0+4*a0*c0-b0^2)/(8*a0*c0-2*b0^2))

[next] [prev] [prev-tail] [front] [up]

4.34.6 y″(x)(a0+b0x+c0x2)+(a1+b1x)y′(x)+2a2y(x)=0

4.34.6 $y^{″} (x) (a0 + b0 x + c0 x^{2}) + (a1 + b1 x) y^{'} (x) + 2 a2 y (x) = 0$