2.17.95 Problems 9401 to 9500

Problem 9401

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime } \cot \left (x \right )+y \sin \left (x \right )^{2}=0} \]
program solution	\[ y = c_{1} \cos \left (\cos \left (x \right )\right )-c_{2} \sin \left (\cot \left (x \right ) {| \sin \left (x \right )|}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sin \left (\cos \left (x \right )\right )+c_{2} \cos \left (\cos \left (x \right )\right ) \]

Problem 9402

ODE	\[ \boxed {y^{\prime \prime }+a y^{\prime } \tan \left (x \right )+y b=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \cos \left (x \right )^{\frac {1}{2}+\frac {a}{2}} \left (c_{1} \operatorname {LegendreP}\left (\frac {\sqrt {a^{2}+4 b}}{2}-\frac {1}{2}, \frac {1}{2}+\frac {a}{2}, \sin \left (x \right )\right )+c_{2} \operatorname {LegendreQ}\left (\frac {\sqrt {a^{2}+4 b}}{2}-\frac {1}{2}, \frac {1}{2}+\frac {a}{2}, \sin \left (x \right )\right )\right ) \]

Problem 9403

ODE	\[ \boxed {y^{\prime \prime }+2 a y^{\prime } \cot \left (x a \right )+\left (-a^{2}+b^{2}\right ) y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{\sqrt {-b^{2}}\, x} \csc \left (x a \right )+\frac {c_{2} \sqrt {-b^{2}}\, \csc \left (x a \right ) {\mathrm e}^{-\sqrt {-b^{2}}\, x}}{2 b^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \csc \left (a x \right ) \left (c_{1} \sin \left (b x \right )+c_{2} \cos \left (b x \right )\right ) \]

Problem 9404

ODE	\[ \boxed {y^{\prime \prime }+a p^{\prime \prime }\left (x \right ) y^{\prime }+\left (a +b p \left (x \right )-4 n a p \left (x \right )^{2}\right ) y=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 9405

ODE	\[ \boxed {y^{\prime \prime }+\frac {\left (11 \operatorname {WeierstrassP}\left (x , a , b\right ) \operatorname {WeierstrassPPrime}\left (x , a , b\right )-6 \operatorname {WeierstrassP}\left (x , a , b\right )^{2}+\frac {a}{2}\right ) y^{\prime }}{\operatorname {WeierstrassPPrime}\left (x , a , b\right )+\operatorname {WeierstrassP}\left (x , a , b\right )^{2}}+\frac {\left (\operatorname {WeierstrassPPrime}\left (x , a , b\right )^{2}-\operatorname {WeierstrassP}\left (x , a , b\right )^{2} \operatorname {WeierstrassPPrime}\left (x , a , b\right )-\operatorname {WeierstrassP}\left (x , a , b\right ) \left (6 \operatorname {WeierstrassP}\left (x , a , b\right )^{2}-\frac {a}{2}\right )\right ) y}{\operatorname {WeierstrassPPrime}\left (x , a , b\right )+\operatorname {WeierstrassP}\left (x , a , b\right )^{2}}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 9406

ODE	\[ \boxed {y^{\prime \prime }+f \left (x \right ) y^{\prime }+g \left (x \right ) y=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 9407

ODE	\[ \boxed {y^{\prime \prime }+f \left (x \right ) y^{\prime }+\left (f^{\prime }\left (x \right )+a \right ) y=g \left (x \right )} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 9408

ODE	\[ \boxed {y^{\prime \prime }+\left (a f \left (x \right )+b \right ) y^{\prime }+\left (c f \left (x \right )+d \right ) y=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 9409

ODE	\[ \boxed {y^{\prime \prime }+f \left (x \right ) y^{\prime }+\left (\frac {f \left (x \right )^{2}}{4}+\frac {f^{\prime }\left (x \right )}{2}+a \right ) y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{x \sqrt {-a}-\frac {\left (\int f \left (x \right )d x \right )}{2}}+c_{2} {\mathrm e}^{x \sqrt {-a}-\frac {\left (\int f \left (x \right )d x \right )}{2}} \left (\int {\mathrm e}^{-2 x \sqrt {-a}}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {\left (\int f \left (x \right )d x \right )}{2}} \left (c_{1} \sinh \left (\sqrt {-a}\, x \right )+c_{2} \cosh \left (\sqrt {-a}\, x \right )\right ) \]

Problem 9410

ODE	\[ \boxed {y^{\prime \prime }-\frac {a f^{\prime }\left (x \right ) y^{\prime }}{f \left (x \right )}+b f \left (x \right )^{2 a} y=0} \]
program solution	\[ y = c_{1} \cos \left (\sqrt {b}\, \left (\int \sqrt {f \left (x \right )^{2 a}}d x \right )\right )+c_{2} \sin \left (\sqrt {b}\, \left (\int \sqrt {f \left (x \right )^{2 a}}d x \right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{i \sqrt {b}\, \left (\int f \left (x \right )^{a}d x \right )}+c_{2} {\mathrm e}^{-i \sqrt {b}\, \left (\int f \left (x \right )^{a}d x \right )} \]

Problem 9411

ODE	\[ \boxed {y^{\prime \prime }-\left (\frac {f^{\prime }\left (x \right )}{f \left (x \right )}+2 a \right ) y^{\prime }+\left (\frac {a f^{\prime }\left (x \right )}{f \left (x \right )}+a^{2}-b^{2} f \left (x \right )^{2}\right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{\int \frac {{\mathrm e}^{2 b \left (\int f \left (x \right )d x \right )} \left (\left (-f \left (x \right ) b +a \right ) {\mathrm e}^{-2 b \left (\int f \left (x \right )d x -c_{1} \right )}-f \left (x \right ) b -a \right )}{-{\mathrm e}^{2 b \left (\int f \left (x \right )d x \right )}+{\mathrm e}^{2 c_{1} b}}d x} c_{2} \]

Problem 9412

ODE	\[ \boxed {y^{\prime \prime }+\frac {f \left (x \right ) f^{\prime \prime \prime }\left (x \right ) y^{\prime }}{f \left (x \right )^{2}+b^{2}}-\frac {a^{2} {f^{\prime }\left (x \right )}^{2} y}{f \left (x \right )^{2}+b^{2}}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 9413

ODE	\[ \boxed {y^{\prime \prime }-\left (\frac {2 f^{\prime }\left (x \right )}{f \left (x \right )}+\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}-\frac {g^{\prime }\left (x \right )}{g \left (x \right )}\right ) y^{\prime }+\left (\frac {f^{\prime }\left (x \right ) \left (\frac {2 f^{\prime }\left (x \right )}{f \left (x \right )}+\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}-\frac {g^{\prime }\left (x \right )}{g \left (x \right )}\right )}{f \left (x \right )}-\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )}-\frac {v^{2} {g^{\prime }\left (x \right )}^{2}}{g \left (x \right )^{2}}+{g^{\prime }\left (x \right )}^{2}\right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = f \left (x \right ) \left (\operatorname {BesselJ}\left (v , g \left (x \right )\right ) c_{1} +\operatorname {BesselY}\left (v , g \left (x \right )\right ) c_{2} \right ) \]

Problem 9414

ODE	\[ \boxed {y^{\prime \prime }-\left (\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}+\frac {\left (2 v -1\right ) g^{\prime }\left (x \right )}{g \left (x \right )}+\frac {2 h^{\prime }\left (x \right )}{h \left (x \right )}\right ) y^{\prime }+\left (\frac {h^{\prime }\left (x \right ) \left (\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}+\frac {\left (2 v -1\right ) g^{\prime }\left (x \right )}{g \left (x \right )}+\frac {2 h^{\prime }\left (x \right )}{h \left (x \right )}\right )}{h \left (x \right )}-\frac {h^{\prime \prime }\left (x \right )}{h \left (x \right )}+{g^{\prime }\left (x \right )}^{2}\right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = g \left (x \right )^{v} h \left (x \right ) \left (\operatorname {BesselJ}\left (v , g \left (x \right )\right ) c_{1} +\operatorname {BesselY}\left (v , g \left (x \right )\right ) c_{2} \right ) \]

Problem 9415

ODE	\[ \boxed {4 y^{\prime \prime }+9 x y=0} \]
program solution	\[ y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\frac {1}{3}, x^{\frac {3}{2}}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\frac {1}{3}, x^{\frac {3}{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {AiryAi}\left (-\frac {3^{\frac {2}{3}} 2^{\frac {1}{3}} x}{2}\right )+c_{2} \operatorname {AiryBi}\left (-\frac {3^{\frac {2}{3}} 2^{\frac {1}{3}} x}{2}\right ) \]

Problem 9416

ODE	\[ \boxed {4 y^{\prime \prime }-\left (x^{2}+a \right ) y=0} \]
program solution	\[ y = -c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (1, 2 \sqrt {x}\right )-c_{2} \sqrt {x}\, \operatorname {BesselY}\left (1, 2 \sqrt {x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} \operatorname {WhittakerW}\left (-\frac {a}{8}, \frac {1}{4}, \frac {x^{2}}{2}\right )+c_{1} \operatorname {WhittakerM}\left (-\frac {a}{8}, \frac {1}{4}, \frac {x^{2}}{2}\right )}{\sqrt {x}} \]

Problem 9417

ODE	\[ \boxed {4 y^{\prime \prime }+4 y^{\prime } \tan \left (x \right )-\left (5 \tan \left (x \right )^{2}+2\right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \frac {i \cos \left (x \right ) \sin \left (x \right ) c_{2} -\ln \left (i \cos \left (x \right )+\sin \left (x \right )\right ) c_{2} +c_{1}}{\sqrt {\cos \left (x \right )}} \]

Problem 9418

ODE	\[ \boxed {a y^{\prime \prime }-\left (a b +c +x \right ) y^{\prime }+\left (b \left (x +c \right )+d \right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{b x} \left (\operatorname {KummerM}\left (-\frac {d}{2}, \frac {1}{2}, \frac {\left (a b -c -x \right )^{2}}{2 a}\right ) c_{1} +\operatorname {KummerU}\left (-\frac {d}{2}, \frac {1}{2}, \frac {\left (a b -c -x \right )^{2}}{2 a}\right ) c_{2} \right ) \]

Problem 9419

ODE	\[ \boxed {a^{2} y^{\prime \prime }+a \left (a^{2}-2 b \,{\mathrm e}^{-x a}\right ) y^{\prime }+b^{2} {\mathrm e}^{-2 x a} y=0} \]
program solution	\[ y = \left (c_{1} {\mathrm e}^{\frac {\sqrt {a^{2}}\, x}{2}}+c_{2} {\mathrm e}^{-\frac {\sqrt {a^{2}}\, x}{2}}\right ) {\mathrm e}^{-\frac {x \,a^{3}+2 b \,{\mathrm e}^{-x a}}{2 a^{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {x \,a^{3}+2 b \,{\mathrm e}^{-a x}}{2 a^{2}}} \left (c_{1} \sinh \left (\frac {a x}{2}\right )+c_{2} \cosh \left (\frac {a x}{2}\right )\right ) \]

Problem 9420

ODE	\[ \boxed {x \left (y^{\prime \prime }+y\right )=\cos \left (x \right )} \]
program solution	\[ y = \cos \left (x \right ) c_{1} +c_{2} \sin \left (x \right )-\frac {\operatorname {Si}\left (2 x \right ) \cos \left (x \right )}{2}+\frac {\left (\ln \left (x \right )+\operatorname {Ci}\left (2 x \right )\right ) \sin \left (x \right )}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\sin \left (x \right ) \operatorname {Ci}\left (2 x \right )}{2}-\frac {\operatorname {Si}\left (2 x \right ) \cos \left (x \right )}{2}+\frac {\left (2 c_{2} +\ln \left (x \right )\right ) \sin \left (x \right )}{2}+\cos \left (x \right ) c_{1} \]

Problem 9421

ODE	\[ \boxed {y^{\prime \prime } x +\left (x +a \right ) y=0} \]
program solution	\[ y = -c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (1, 2 \sqrt {x}\right )-c_{2} \sqrt {x}\, \operatorname {BesselY}\left (1, 2 \sqrt {x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {WhittakerM}\left (-\frac {i a}{2}, \frac {1}{2}, 2 i x \right )+c_{2} \operatorname {WhittakerW}\left (-\frac {i a}{2}, \frac {1}{2}, 2 i x \right ) \]

Problem 9422

ODE	\[ \boxed {y^{\prime \prime } x +y^{\prime }=0} \]
program solution	\[ y = \ln \left (x \right ) c_{1} +c_{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \ln \left (x \right ) c_{2} +c_{1} \]

Problem 9423

ODE	\[ \boxed {y^{\prime \prime } x +y^{\prime }+a y=0} \]
program solution	\[ y = c_{1} \operatorname {BesselJ}\left (0, 2 \sqrt {a}\, \sqrt {x}\right )+c_{2} \operatorname {BesselY}\left (0, 2 \sqrt {a}\, \sqrt {x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {BesselJ}\left (0, 2 \sqrt {a}\, \sqrt {x}\right )+c_{2} \operatorname {BesselY}\left (0, 2 \sqrt {a}\, \sqrt {x}\right ) \]

Problem 9424

ODE	\[ \boxed {y^{\prime \prime } x +y^{\prime }+l x y=0} \]
program solution	\[ y = c_{1} \operatorname {BesselJ}\left (0, x \sqrt {l}\right )+c_{2} \operatorname {BesselY}\left (0, x \sqrt {l}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {BesselJ}\left (0, \sqrt {l}\, x \right )+c_{2} \operatorname {BesselY}\left (0, \sqrt {l}\, x \right ) \]

Problem 9425

ODE	\[ \boxed {y^{\prime \prime } x +y^{\prime }+\left (x +a \right ) y=0} \]
program solution	\[ y = \operatorname {BesselJ}\left (0, 2 \sqrt {x}\right ) c_{1} +\operatorname {BesselY}\left (0, 2 \sqrt {x}\right ) c_{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-i x} \left (\operatorname {KummerM}\left (\frac {1}{2}+\frac {i a}{2}, 1, 2 i x \right ) c_{1} +\operatorname {KummerU}\left (\frac {1}{2}+\frac {i a}{2}, 1, 2 i x \right ) c_{2} \right ) \]

Problem 9426

ODE	\[ \boxed {y^{\prime \prime } x -y^{\prime }+a y=0} \]
program solution	\[ y = c_{1} x \operatorname {BesselJ}\left (2, 2 \sqrt {a}\, \sqrt {x}\right )+c_{2} x \operatorname {BesselY}\left (2, 2 \sqrt {a}\, \sqrt {x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\operatorname {BesselJ}\left (1, 2 \sqrt {a}\, \sqrt {x}\right ) \sqrt {x}\, c_{1} +\operatorname {BesselY}\left (1, 2 \sqrt {a}\, \sqrt {x}\right ) \sqrt {x}\, c_{2} -\sqrt {a}\, x \left (c_{1} \operatorname {BesselJ}\left (0, 2 \sqrt {a}\, \sqrt {x}\right )+c_{2} \operatorname {BesselY}\left (0, 2 \sqrt {a}\, \sqrt {x}\right )\right )}{\sqrt {a}} \]

Problem 9427

ODE	\[ \boxed {y^{\prime \prime } x -y^{\prime }-y a \,x^{3}=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {\sqrt {a}\, x^{2}}{2}}+\frac {c_{2} {\mathrm e}^{\frac {\sqrt {a}\, x^{2}}{2}}}{2 \sqrt {a}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sinh \left (\frac {x^{2} \sqrt {a}}{2}\right )+c_{2} \cosh \left (\frac {x^{2} \sqrt {a}}{2}\right ) \]

Problem 9428

ODE	\[ \boxed {y^{\prime \prime } x -y^{\prime }+x^{3} \left ({\mathrm e}^{x^{2}}-v^{2}\right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {BesselJ}\left (v , {\mathrm e}^{\frac {x^{2}}{2}}\right )+c_{2} \operatorname {BesselY}\left (v , {\mathrm e}^{\frac {x^{2}}{2}}\right ) \]

Problem 9429

ODE	\[ \boxed {y^{\prime \prime } x +2 y^{\prime }-x y={\mathrm e}^{x}} \]
program solution	\[ y = \frac {2 c_{1} {\mathrm e}^{-x}+c_{2} {\mathrm e}^{x}}{2 x}+\frac {{\mathrm e}^{x} \left (2 x -1\right )}{4 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {2 \sinh \left (x \right ) c_{2} +2 \cosh \left (x \right ) c_{1} +x \,{\mathrm e}^{x}}{2 x} \]

Problem 9430

ODE	\[ \boxed {y^{\prime \prime } x +2 y^{\prime }+y a x=0} \]
program solution	\[ y = \frac {c_{1} {\mathrm e}^{x \sqrt {-a}}}{x}-\frac {c_{2} {\mathrm e}^{-x \sqrt {-a}}}{2 x \sqrt {-a}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sinh \left (\sqrt {-a}\, x \right )+c_{2} \cosh \left (\sqrt {-a}\, x \right )}{x} \]

Problem 9431

ODE	\[ \boxed {y^{\prime \prime } x +2 y^{\prime }+a \,x^{2} y=0} \]
program solution	\[ y = \frac {c_{1} \operatorname {BesselJ}\left (\frac {1}{3}, \frac {2 \sqrt {a}\, x^{\frac {3}{2}}}{3}\right )}{\sqrt {x}}+\frac {c_{2} \operatorname {BesselY}\left (\frac {1}{3}, \frac {2 \sqrt {a}\, x^{\frac {3}{2}}}{3}\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \operatorname {BesselJ}\left (\frac {1}{3}, \frac {2 \sqrt {a}\, x^{\frac {3}{2}}}{3}\right )+c_{2} \operatorname {BesselY}\left (\frac {1}{3}, \frac {2 \sqrt {a}\, x^{\frac {3}{2}}}{3}\right )}{\sqrt {x}} \]

Problem 9432

ODE	\[ \boxed {y^{\prime \prime } x -2 y^{\prime }+a y=0} \]
program solution	\[ y = c_{1} x^{\frac {3}{2}} \operatorname {BesselJ}\left (3, 2 \sqrt {a}\, \sqrt {x}\right )+c_{2} x^{\frac {3}{2}} \operatorname {BesselY}\left (3, 2 \sqrt {a}\, \sqrt {x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\sqrt {x}\, \left (c_{1} \left (a x -2\right ) \operatorname {BesselJ}\left (1, 2 \sqrt {a}\, \sqrt {x}\right )+c_{2} \left (a x -2\right ) \operatorname {BesselY}\left (1, 2 \sqrt {a}\, \sqrt {x}\right )+2 \sqrt {a}\, \sqrt {x}\, \left (c_{1} \operatorname {BesselJ}\left (0, 2 \sqrt {a}\, \sqrt {x}\right )+c_{2} \operatorname {BesselY}\left (0, 2 \sqrt {a}\, \sqrt {x}\right )\right )\right )}{a} \]

Problem 9433

ODE	\[ \boxed {y^{\prime \prime } x +v y^{\prime }+a y=0} \]
program solution	\[ y = c_{1} x^{\frac {1}{2}-\frac {v}{2}} \operatorname {BesselJ}\left (-v +1, 2 \sqrt {a}\, \sqrt {x}\right )+c_{2} x^{\frac {1}{2}-\frac {v}{2}} \operatorname {BesselY}\left (-v +1, 2 \sqrt {a}\, \sqrt {x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-\sqrt {a}\, \sqrt {x}\, \operatorname {BesselJ}\left (v +1, 2 \sqrt {a}\, \sqrt {x}\right ) c_{1} -\sqrt {a}\, \sqrt {x}\, \operatorname {BesselY}\left (v +1, 2 \sqrt {a}\, \sqrt {x}\right ) c_{2} +v \left (\operatorname {BesselJ}\left (v , 2 \sqrt {a}\, \sqrt {x}\right ) c_{1} +\operatorname {BesselY}\left (v , 2 \sqrt {a}\, \sqrt {x}\right ) c_{2} \right )\right ) x^{-\frac {v}{2}}}{\sqrt {a}} \]

Problem 9434

ODE	\[ \boxed {y^{\prime \prime } x +a y^{\prime }+y b x=0} \]
program solution	\[ y = c_{1} x^{-\frac {a}{2}+\frac {1}{2}} \operatorname {BesselJ}\left (-\frac {a}{2}+\frac {1}{2}, x \sqrt {b}\right )+c_{2} x^{-\frac {a}{2}+\frac {1}{2}} \operatorname {BesselY}\left (-\frac {a}{2}+\frac {1}{2}, x \sqrt {b}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\operatorname {BesselY}\left (\frac {a}{2}-\frac {1}{2}, \sqrt {b}\, x \right ) c_{2} +\operatorname {BesselJ}\left (\frac {a}{2}-\frac {1}{2}, \sqrt {b}\, x \right ) c_{1} \right ) x^{-\frac {a}{2}+\frac {1}{2}} \]

Problem 9435

ODE	\[ \boxed {y^{\prime \prime } x +a y^{\prime }+b \,x^{\operatorname {a1}} y=0} \]
program solution	\[ y = c_{1} x^{-\frac {a}{2}+\frac {1}{2}} \operatorname {BesselJ}\left (-\frac {-1+a}{1+\operatorname {a1}}, \frac {2 \sqrt {b}\, x^{\frac {1}{2}+\frac {\operatorname {a1}}{2}}}{1+\operatorname {a1}}\right )+c_{2} x^{-\frac {a}{2}+\frac {1}{2}} \operatorname {BesselY}\left (-\frac {-1+a}{1+\operatorname {a1}}, \frac {2 \sqrt {b}\, x^{\frac {1}{2}+\frac {\operatorname {a1}}{2}}}{1+\operatorname {a1}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\operatorname {BesselY}\left (\frac {a -1}{\operatorname {a1} +1}, \frac {2 \sqrt {b}\, x^{\frac {\operatorname {a1}}{2}+\frac {1}{2}}}{\operatorname {a1} +1}\right ) c_{2} +\operatorname {BesselJ}\left (\frac {a -1}{\operatorname {a1} +1}, \frac {2 \sqrt {b}\, x^{\frac {\operatorname {a1}}{2}+\frac {1}{2}}}{\operatorname {a1} +1}\right ) c_{1} \right ) x^{-\frac {a}{2}+\frac {1}{2}} \]

Problem 9436

ODE	\[ \boxed {y^{\prime \prime } x +\left (x +b \right ) y^{\prime }+a y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-x} \left (\operatorname {KummerM}\left (-a +b , b , x\right ) c_{1} +\operatorname {KummerU}\left (-a +b , b , x\right ) c_{2} \right ) \]

Problem 9437

ODE	\[ \boxed {y^{\prime \prime } x +\left (x +a +b \right ) y^{\prime }+a y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-x} \left (\operatorname {KummerU}\left (b , a +b , x\right ) c_{2} +\operatorname {KummerM}\left (b , a +b , x\right ) c_{1} \right ) \]

Problem 9438

ODE	\[ \boxed {y^{\prime \prime } x -y^{\prime } x -y=x \left (x +1\right ) {\mathrm e}^{x}} \]
program solution	\[ y = -\ln \left (-x \right ) x \,{\mathrm e}^{x}+\operatorname {expIntegral}_{1}\left (x \right ) c_{1} x \,{\mathrm e}^{x}+\left (c_{2} x +x^{2}-1\right ) {\mathrm e}^{x}-c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {expIntegral}_{1}\left (x \right ) {\mathrm e}^{x} c_{1} x +\left (x^{2}+x c_{2} -x \ln \left (x \right )-1\right ) {\mathrm e}^{x}-c_{1} \]

Problem 9439

ODE	\[ \boxed {y^{\prime \prime } x -y^{\prime } x -a y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = x \left (\operatorname {KummerM}\left (a +1, 2, x\right ) c_{1} +\operatorname {KummerU}\left (a +1, 2, x\right ) c_{2} \right ) \]

Problem 9440

ODE	\[ \boxed {y^{\prime \prime } x -\left (x +1\right ) y^{\prime }+y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{x}+\left (-x -1\right ) c_{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x} c_{2} +c_{1} x +c_{1} \]

Problem 9441

ODE	\[ \boxed {y^{\prime \prime } x -\left (x +1\right ) y^{\prime }-2 \left (x -1\right ) y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{2 x}-\frac {c_{2} \left (3 x +1\right ) {\mathrm e}^{-x}}{9} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{2 x} c_{1} +c_{2} {\mathrm e}^{-x} \left (3 x +1\right ) \]

Problem 9442

ODE	\[ \boxed {y^{\prime \prime } x +\left (b -x \right ) y^{\prime }-a y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {KummerM}\left (a , b , x\right )+c_{2} \operatorname {KummerU}\left (a , b , x\right ) \]

Problem 9443

ODE	\[ \boxed {y^{\prime \prime } x -2 \left (x -1\right ) y^{\prime }-y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \left (\operatorname {BesselK}\left (0, -x \right ) c_{2} -\operatorname {BesselK}\left (1, -x \right ) c_{2} +c_{1} \left (\operatorname {BesselI}\left (0, -x \right )+\operatorname {BesselI}\left (1, -x \right )\right )\right ) {\mathrm e}^{x} \]

Problem 9444

ODE	\[ \boxed {y^{\prime \prime } x -\left (3 x -2\right ) y^{\prime }-\left (2 x -3\right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {x \left (-3+\sqrt {17}\right )}{2}} \left (\operatorname {KummerM}\left (1-\frac {6 \sqrt {17}}{17}, 2, \sqrt {17}\, x \right ) c_{1} +\operatorname {KummerU}\left (1-\frac {6 \sqrt {17}}{17}, 2, \sqrt {17}\, x \right ) c_{2} \right ) \]

Problem 9445

ODE	\[ \boxed {y^{\prime \prime } x +\left (x a +b +n \right ) y^{\prime }+n a y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-a x} \left (\operatorname {KummerU}\left (b , b +n , a x \right ) c_{2} +\operatorname {KummerM}\left (b , b +n , a x \right ) c_{1} \right ) \]

Problem 9446

ODE	\[ \boxed {y^{\prime \prime } x -\left (a +b \right ) \left (x +1\right ) y^{\prime }+a b x y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{b x} x^{b +a +1} \left (\operatorname {KummerM}\left (\frac {a^{2}+a b +a -b}{a -b}, b +2+a , x \left (a -b \right )\right ) c_{1} +\operatorname {KummerU}\left (\frac {a^{2}+a b +a -b}{a -b}, b +2+a , x \left (a -b \right )\right ) c_{2} \right ) \]

Problem 9447

ODE	\[ \boxed {y^{\prime \prime } x +\left (x \left (a +b \right )+m +n \right ) y^{\prime }+\left (a b x +a n +b m \right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-a x} \left (\operatorname {KummerM}\left (m , m +n , x \left (a -b \right )\right ) c_{1} +\operatorname {KummerU}\left (m , m +n , x \left (a -b \right )\right ) c_{2} \right ) \]

Problem 9448

ODE	\[ \boxed {y^{\prime \prime } x -2 \left (x a +b \right ) y^{\prime }+\left (a^{2} x +2 a b \right ) y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{x a}+\frac {c_{2} {\mathrm e}^{x a} x^{2 b +1}}{2 b +1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{a x} \left (c_{1} +x^{2 b +1} c_{2} \right ) \]

Problem 9449

ODE	\[ \boxed {y^{\prime \prime } x +\left (x a +b \right ) y^{\prime }+\left (c x +d \right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {x \left (\sqrt {a^{2}-4 c}+a \right )}{2}} \left (\operatorname {KummerU}\left (\frac {b \sqrt {a^{2}-4 c}+a b -2 d}{2 \sqrt {a^{2}-4 c}}, b , \sqrt {a^{2}-4 c}\, x \right ) c_{2} +\operatorname {KummerM}\left (\frac {b \sqrt {a^{2}-4 c}+a b -2 d}{2 \sqrt {a^{2}-4 c}}, b , \sqrt {a^{2}-4 c}\, x \right ) c_{1} \right ) \]

Problem 9450

ODE	\[ \boxed {y^{\prime \prime } x -\left (x^{2}-x \right ) y^{\prime }+\left (x -1\right ) y=0} \]
program solution	\[ y = c_{1} x +c_{2} x \left (\int \frac {{\mathrm e}^{\frac {x \left (x -2\right )}{2}}}{x^{2}}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (\int \frac {{\mathrm e}^{\frac {x \left (x -2\right )}{2}}}{x^{2}}d x \right ) c_{1} +c_{2} \right ) x \]

Problem 9451

ODE	\[ \boxed {y^{\prime \prime } x -\left (x^{2}-x -2\right ) y^{\prime }-x \left (x +3\right ) y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{\frac {x^{2}}{2}}+c_{2} {\mathrm e}^{\frac {x^{2}}{2}} \left (\int \frac {{\mathrm e}^{-\frac {x \left (x +2\right )}{2}}}{x^{2}}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{\frac {x^{2}}{2}} \left (c_{1} +\left (\int \frac {{\mathrm e}^{-\frac {x \left (x +2\right )}{2}}}{x^{2}}d x \right ) c_{2} \right ) \]

Problem 9452

ODE	\[ \boxed {y^{\prime \prime } x -\left (2 x^{2} a +1\right ) y^{\prime }+b \,x^{3} y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {x^{2} \left (-a +\sqrt {a^{2}-b}\right )}{2}}+\frac {c_{2} {\mathrm e}^{\frac {x^{2} \left (a +\sqrt {a^{2}-b}\right )}{2}}}{2 \sqrt {a^{2}-b}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {x^{2} \left (\sqrt {a^{2}-b}+a \right )}{2}}+c_{2} {\mathrm e}^{-\frac {x^{2} \left (\sqrt {a^{2}-b}-a \right )}{2}} \]

Problem 9453

ODE	\[ \boxed {y^{\prime \prime } x -2 \left (x^{2}-a \right ) y^{\prime }+2 n x y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {KummerM}\left (-\frac {n}{2}, \frac {1}{2}+a , x^{2}\right )+c_{2} \operatorname {KummerU}\left (-\frac {n}{2}, \frac {1}{2}+a , x^{2}\right ) \]

Problem 9454

ODE	\[ \boxed {y^{\prime \prime } x +\left (4 x^{2}-1\right ) y^{\prime }-4 y x^{3}=4 x^{5}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x^{2} \left (1+\sqrt {2}\right )}+\frac {c_{2} \sqrt {2}\, {\mathrm e}^{x^{2} \left (\sqrt {2}-1\right )}}{8}-x^{2}-2 \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x^{2} \left (\sqrt {2}-1\right )} c_{2} +{\mathrm e}^{-x^{2} \left (1+\sqrt {2}\right )} c_{1} -x^{2}-2 \]

Problem 9455

ODE	\[ \boxed {y^{\prime \prime } x +\left (2 a \,x^{3}-1\right ) y^{\prime }+\left (a^{2} x^{3}+a \right ) x^{2} y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {a \,x^{3}}{3}}+\frac {c_{2} {\mathrm e}^{-\frac {a \,x^{3}}{3}} x^{2}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {x^{3} a}{3}} \left (c_{2} x^{2}+c_{1} \right ) \]

Problem 9456

ODE	\[ \boxed {y^{\prime \prime } x +\left (2 a x \ln \left (x \right )+1\right ) y^{\prime }+\left (a^{2} x \ln \left (x \right )^{2}+a \ln \left (x \right )+a \right ) y=0} \]
program solution	\[ y = c_{1} x^{-x a} {\mathrm e}^{x a}+c_{2} x^{-x a} {\mathrm e}^{x a} \ln \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{-a x} {\mathrm e}^{a x} \left (\ln \left (x \right ) c_{2} +c_{1} \right ) \]

Problem 9457

ODE	\[ \boxed {y^{\prime \prime } x +\left (x f \left (x \right )+2\right ) y^{\prime }+f \left (x \right ) y=0} \]
program solution	\[ y = \frac {\int c_{1} {\mathrm e}^{\int -f \left (x \right )d x}d x +c_{2}}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} \left (\int {\mathrm e}^{-\left (\int \frac {f \left (x \right ) x +2}{x}d x \right )} x^{2}d x \right )+c_{1}}{x} \]

Problem 9458

ODE	\[ \boxed {\left (x -3\right ) y^{\prime \prime }-\left (4 x -9\right ) y^{\prime }+\left (3 x -6\right ) y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{x}+\frac {c_{2} {\mathrm e}^{3 x} \left (4 x^{3}-42 x^{2}+150 x -183\right )}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 4 c_{2} \left (x^{3}-\frac {21}{2} x^{2}+\frac {75}{2} x -\frac {183}{4}\right ) {\mathrm e}^{3 x}+c_{1} {\mathrm e}^{x} \]

Problem 9459

ODE	\[ \boxed {2 y^{\prime \prime } x +y^{\prime }+a y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{\sqrt {2}\, \sqrt {-x a}}-\frac {c_{2} \sqrt {2}\, \sqrt {-x a}\, \left ({\mathrm e}^{\sqrt {2}\, \sqrt {-x a}}-{\mathrm e}^{-\sqrt {2}\, \sqrt {-x a}}\right )}{2 a \sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sin \left (\sqrt {x}\, \sqrt {2}\, \sqrt {a}\right )+c_{2} \cos \left (\sqrt {x}\, \sqrt {2}\, \sqrt {a}\right ) \]

Problem 9460

ODE	\[ \boxed {2 y^{\prime \prime } x -\left (x -1\right ) y^{\prime }+a y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (\operatorname {KummerM}\left (-a +\frac {1}{2}, \frac {3}{2}, \frac {x}{2}\right ) c_{1} +\operatorname {KummerU}\left (-a +\frac {1}{2}, \frac {3}{2}, \frac {x}{2}\right ) c_{2} \right ) \]

Problem 9461

ODE	\[ \boxed {2 y^{\prime \prime } x -\left (2 x -1\right ) y^{\prime }+a y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (\operatorname {KummerM}\left (-\frac {a}{2}+\frac {1}{2}, \frac {3}{2}, x\right ) c_{1} +\operatorname {KummerU}\left (-\frac {a}{2}+\frac {1}{2}, \frac {3}{2}, x\right ) c_{2} \right ) \]

Problem 9462

ODE	\[ \boxed {\left (2 x -1\right ) y^{\prime \prime }-\left (3 x -4\right ) y^{\prime }+\left (x -3\right ) y=0} \]
program solution	\[ y = \frac {c_{1} 2^{\frac {3}{8}} {\mathrm e}^{x}}{2}+c_{2} 2^{\frac {5}{8}} {\mathrm e}^{x} \left (\int \frac {{\mathrm e}^{-\frac {x}{2}}}{\left (2 x -1\right )^{\frac {5}{4}}}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{-\frac {1}{4}+x} \sqrt {2}\, \left (\left (\frac {c_{1}}{4}+c_{2} \right ) \Gamma \left (-\frac {1}{4}, -\frac {1}{4}+\frac {x}{2}\right )+\Gamma \left (\frac {3}{4}\right ) c_{1} \right )}{2} \]

Problem 9463

ODE	\[ \boxed {4 y^{\prime \prime } x -\left (x +a \right ) y=0} \]
program solution	\[ y = -c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (1, 2 \sqrt {x}\right )-c_{2} \sqrt {x}\, \operatorname {BesselY}\left (1, 2 \sqrt {x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {WhittakerM}\left (-\frac {a}{4}, \frac {1}{2}, x\right )+c_{2} \operatorname {WhittakerW}\left (-\frac {a}{4}, \frac {1}{2}, x\right ) \]

Problem 9464

ODE	\[ \boxed {4 y^{\prime \prime } x +2 y^{\prime }-y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{\sqrt {x}}-c_{2} {\mathrm e}^{-\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sinh \left (\sqrt {x}\right )+c_{2} \cosh \left (\sqrt {x}\right ) \]

Problem 9465

ODE	\[ \boxed {4 y^{\prime \prime } x +4 y^{\prime }-\left (x +2\right ) y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{\frac {x}{2}}-c_{2} {\mathrm e}^{\frac {x}{2}} \operatorname {expIntegral}_{1}\left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{\frac {x}{2}} \left (c_{1} +\operatorname {expIntegral}_{1}\left (x \right ) c_{2} \right ) \]

Problem 9466

ODE	\[ \boxed {4 y^{\prime \prime } x +4 y-\left (x +2\right ) y+l y=0} \]
program solution	\[ y = -c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (1, 2 \sqrt {x}\right )-c_{2} \sqrt {x}\, \operatorname {BesselY}\left (1, 2 \sqrt {x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {WhittakerM}\left (\frac {l}{4}+\frac {1}{2}, \frac {1}{2}, x\right )+c_{2} \operatorname {WhittakerW}\left (\frac {l}{4}+\frac {1}{2}, \frac {1}{2}, x\right ) \]

Problem 9467

ODE	\[ \boxed {4 y^{\prime \prime } x +4 m y^{\prime }-\left (x -2 m -4 n \right ) y=0} \]
program solution	\[ y = c_{1} x^{\frac {1}{2}-\frac {m}{2}} \operatorname {BesselJ}\left (-m +1, 2 \sqrt {x}\right )+c_{2} x^{\frac {1}{2}-\frac {m}{2}} \operatorname {BesselY}\left (-m +1, 2 \sqrt {x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {x}{2}} \left (\operatorname {KummerM}\left (-n , m , x\right ) c_{1} +\operatorname {KummerU}\left (-n , m , x\right ) c_{2} \right ) \]

Problem 9468

ODE	\[ \boxed {16 y^{\prime \prime } x +8 y^{\prime }-\left (x +a \right ) y=0} \]
program solution	\[ y = \frac {c_{1} \cos \left (2 \sqrt {x}\right )}{\sqrt {\pi }}+\frac {c_{2} \sin \left (2 \sqrt {x}\right )}{\sqrt {\pi }} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, {\mathrm e}^{-\frac {x}{4}} \left (\operatorname {KummerU}\left (\frac {a}{8}+\frac {3}{4}, \frac {3}{2}, \frac {x}{2}\right ) c_{2} +\operatorname {KummerM}\left (\frac {a}{8}+\frac {3}{4}, \frac {3}{2}, \frac {x}{2}\right ) c_{1} \right ) \]

Problem 9469

ODE	\[ \boxed {a x y^{\prime \prime }+b y^{\prime }+y c=0} \]
program solution	\[ y = c_{1} x^{-\frac {b}{2 a}+\frac {1}{2}} \operatorname {BesselJ}\left (\frac {a -b}{a}, \frac {2 \sqrt {c a}\, \sqrt {x}}{a}\right )+c_{2} x^{-\frac {b}{2 a}+\frac {1}{2}} \operatorname {BesselY}\left (\frac {a -b}{a}, \frac {2 \sqrt {c a}\, \sqrt {x}}{a}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{-\frac {b}{2 a}} \left (-\sqrt {x}\, \operatorname {BesselJ}\left (\frac {a +b}{a}, 2 \sqrt {\frac {c}{a}}\, \sqrt {x}\right ) \sqrt {\frac {c}{a}}\, c_{1} a -\sqrt {x}\, \operatorname {BesselY}\left (\frac {a +b}{a}, 2 \sqrt {\frac {c}{a}}\, \sqrt {x}\right ) \sqrt {\frac {c}{a}}\, c_{2} a +\operatorname {BesselJ}\left (\frac {b}{a}, 2 \sqrt {\frac {c}{a}}\, \sqrt {x}\right ) c_{1} b +\operatorname {BesselY}\left (\frac {b}{a}, 2 \sqrt {\frac {c}{a}}\, \sqrt {x}\right ) c_{2} b \right )}{a \sqrt {\frac {c}{a}}} \]

Problem 9470

ODE	\[ \boxed {a x y^{\prime \prime }+\left (b x +3 a \right ) y^{\prime }+3 y b=0} \]
program solution	\[ y = c_{3} \left ({\mathrm e}^{-\frac {x b}{a}} \operatorname {expIntegral}_{1}\left (-\frac {x b}{a}\right ) c_{1} b^{2}+\frac {c_{1} a b}{x}-2 \,{\mathrm e}^{-\frac {x b}{a}} c_{2} a^{2}+\frac {c_{1} a^{2}}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{-\frac {b x}{a}} \operatorname {expIntegral}_{1}\left (-\frac {b x}{a}\right ) c_{2} b^{2} x^{2}+c_{1} {\mathrm e}^{-\frac {b x}{a}} x^{2}+c_{2} a \left (b x +a \right )}{x^{2}} \]

Problem 9471

ODE	\[ \boxed {5 \left (x a +b \right ) y^{\prime \prime }+8 a y^{\prime }+c \left (x a +b \right )^{\frac {1}{5}} y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sinh \left (\frac {\left (a x +b \right )^{\frac {3}{5}} \sqrt {5}\, \sqrt {-c}}{3 a}\right )+c_{2} \cosh \left (\frac {\left (a x +b \right )^{\frac {3}{5}} \sqrt {5}\, \sqrt {-c}}{3 a}\right )}{\left (a x +b \right )^{\frac {3}{5}}} \]

Problem 9472

ODE	\[ \boxed {2 a x y^{\prime \prime }+\left (b x +a \right ) y^{\prime }+y c=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, {\mathrm e}^{-\frac {b x}{2 a}} \left (\operatorname {KummerU}\left (\frac {b -c}{b}, \frac {3}{2}, \frac {b x}{2 a}\right ) c_{2} +\operatorname {KummerM}\left (\frac {b -c}{b}, \frac {3}{2}, \frac {b x}{2 a}\right ) c_{1} \right ) \]

Problem 9473

ODE	\[ \boxed {2 a x y^{\prime \prime }+\left (b x +3 a \right ) y^{\prime }+y c=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \frac {\left (-c_{1} c \left (a \left (b +4 c \right )-b^{2} x \right ) \operatorname {KummerM}\left (\frac {b -2 c}{2 b}, \frac {3}{2}, \frac {b x}{2 a}\right )+c_{2} b \left (a \left (b +4 c \right )-b^{2} x \right ) \operatorname {KummerU}\left (\frac {b -2 c}{2 b}, \frac {3}{2}, \frac {b x}{2 a}\right )+2 a \left (c c_{1} \left (c +b \right ) \operatorname {KummerM}\left (-\frac {b +2 c}{2 b}, \frac {3}{2}, \frac {b x}{2 a}\right )+\operatorname {KummerU}\left (-\frac {b +2 c}{2 b}, \frac {3}{2}, \frac {b x}{2 a}\right ) b^{2} c_{2} \right )\right ) {\mathrm e}^{-\frac {b x}{2 a}}}{a \left (b -2 c \right ) c} \]

Problem 9474

ODE	\[ \boxed {\left (\operatorname {a2} x +\operatorname {b2} \right ) y^{\prime \prime }+\left (\operatorname {a1} x +\operatorname {b1} \right ) y^{\prime }+\left (\operatorname {a0} x +\operatorname {b0} \right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {\left (\sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}+\operatorname {a1} \right ) x}{2 \operatorname {a2}}} \left (\operatorname {a2} x +\operatorname {b2} \right )^{\frac {\operatorname {a1} \operatorname {b2} +\operatorname {a2}^{2}-\operatorname {a2} \operatorname {b1}}{\operatorname {a2}^{2}}} \left (\operatorname {KummerU}\left (\frac {\left (\operatorname {a1} \operatorname {b2} +2 \operatorname {a2}^{2}-\operatorname {a2} \operatorname {b1} \right ) \sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}-2 \operatorname {a2}^{2} \operatorname {b0} +\left (2 \operatorname {a0} \operatorname {b2} +\operatorname {a1} \operatorname {b1} \right ) \operatorname {a2} -\operatorname {a1}^{2} \operatorname {b2}}{2 \sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, \operatorname {a2}^{2}}, \frac {\operatorname {a1} \operatorname {b2} +2 \operatorname {a2}^{2}-\operatorname {a2} \operatorname {b1}}{\operatorname {a2}^{2}}, \frac {\sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, \left (\operatorname {a2} x +\operatorname {b2} \right )}{\operatorname {a2}^{2}}\right ) c_{2} +\operatorname {KummerM}\left (\frac {\left (\operatorname {a1} \operatorname {b2} +2 \operatorname {a2}^{2}-\operatorname {a2} \operatorname {b1} \right ) \sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}-2 \operatorname {a2}^{2} \operatorname {b0} +\left (2 \operatorname {a0} \operatorname {b2} +\operatorname {a1} \operatorname {b1} \right ) \operatorname {a2} -\operatorname {a1}^{2} \operatorname {b2}}{2 \sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, \operatorname {a2}^{2}}, \frac {\operatorname {a1} \operatorname {b2} +2 \operatorname {a2}^{2}-\operatorname {a2} \operatorname {b1}}{\operatorname {a2}^{2}}, \frac {\sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, \left (\operatorname {a2} x +\operatorname {b2} \right )}{\operatorname {a2}^{2}}\right ) c_{1} \right ) \]

Problem 9475

ODE	\[ \boxed {x^{2} y^{\prime \prime }-6 y=0} \]
program solution	\[ y = \frac {c_{1}}{x^{2}}+\frac {c_{2} x^{3}}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{5}+c_{2}}{x^{2}} \]

Problem 9476

ODE	\[ \boxed {x^{2} y^{\prime \prime }-12 y=0} \]
program solution	\[ y = \frac {c_{1}}{x^{3}}+\frac {c_{2} x^{4}}{7} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{7}+c_{1}}{x^{3}} \]

Problem 9477

ODE	\[ \boxed {x^{2} y^{\prime \prime }+a y=0} \]
program solution	\[ y = c_{1} x^{\frac {1}{2}+\frac {\sqrt {1-4 a}}{2}}-\frac {c_{2} x^{\frac {1}{2}-\frac {\sqrt {1-4 a}}{2}}}{\sqrt {1-4 a}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (c_{1} x^{\frac {\sqrt {-4 a +1}}{2}}+c_{2} x^{-\frac {\sqrt {-4 a +1}}{2}}\right ) \]

Problem 9478

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (x a +b \right ) y=0} \]
program solution	\[ y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\sqrt {-4 b +1}, 2 \sqrt {a}\, \sqrt {x}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\sqrt {-4 b +1}, 2 \sqrt {a}\, \sqrt {x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\operatorname {BesselY}\left (\sqrt {1-4 b}, 2 \sqrt {a}\, \sqrt {x}\right ) c_{2} +\operatorname {BesselJ}\left (\sqrt {1-4 b}, 2 \sqrt {a}\, \sqrt {x}\right ) c_{1} \right ) \sqrt {x} \]

Problem 9479

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (x^{2}-2\right ) y=0} \]
program solution	\[ y = \frac {c_{1} \left (x -i\right ) {\mathrm e}^{-i x}}{x}-\frac {c_{2} \left (i x -1\right ) {\mathrm e}^{i x}}{2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{1} x +c_{2} \right ) \cos \left (x \right )+\sin \left (x \right ) \left (x c_{2} -c_{1} \right )}{x} \]

Problem 9480

ODE	\[ \boxed {x^{2} y^{\prime \prime }-\left (x^{2} a +2\right ) y=0} \]
program solution	\[ y = \frac {c_{1} \left (x \sqrt {a}+1\right ) {\mathrm e}^{-x \sqrt {a}}}{x \sqrt {a}}+\frac {c_{2} \left (x \sqrt {a}-1\right ) \left (x a +\sqrt {a}\right ) {\mathrm e}^{x \sqrt {a}}}{2 x \,a^{\frac {3}{2}} \left (x \sqrt {a}+1\right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} {\mathrm e}^{-x \sqrt {a}} \left (a x +\sqrt {a}\right )+c_{1} {\mathrm e}^{x \sqrt {a}} \left (a x -\sqrt {a}\right )}{x} \]

Problem 9481

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (x^{2} a^{2}-6\right ) y=0} \]
program solution	\[ y = \frac {c_{1} \left (x^{2} a^{2}-3 i x a -3\right ) {\mathrm e}^{-i x a}}{x^{2} a^{2}}-\frac {c_{2} {\mathrm e}^{i x a} \left (i x^{2} a^{2}-3 x a -3 i\right )}{2 a^{3} x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (a^{2} c_{1} x^{2}+3 a c_{2} x -3 c_{1} \right ) \cos \left (a x \right )+\sin \left (a x \right ) \left (a^{2} c_{2} x^{2}-3 a c_{1} x -3 c_{2} \right )}{x^{2}} \]

Problem 9482

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (x^{2} a -v \left (v -1\right )\right ) y=0} \]
program solution	\[ y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (-v +\frac {1}{2}, x \sqrt {a}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (-v +\frac {1}{2}, x \sqrt {a}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\operatorname {BesselJ}\left (v -\frac {1}{2}, x \sqrt {a}\right ) c_{1} +\operatorname {BesselY}\left (v -\frac {1}{2}, x \sqrt {a}\right ) c_{2} \right ) \sqrt {x} \]

Problem 9483

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (x^{2} a +b x +c \right ) y=0} \]
program solution	\[ y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\sqrt {-4 c +1}, 2 \sqrt {x}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\sqrt {-4 c +1}, 2 \sqrt {x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {WhittakerM}\left (-\frac {i b}{2 \sqrt {a}}, \frac {\sqrt {1-4 c}}{2}, 2 i x \sqrt {a}\right )+c_{2} \operatorname {WhittakerW}\left (-\frac {i b}{2 \sqrt {a}}, \frac {\sqrt {1-4 c}}{2}, 2 i x \sqrt {a}\right ) \]

Problem 9484

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (a \,x^{k}-b \left (-1+b \right )\right ) y=0} \]
program solution	\[ y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (-\frac {-1+2 b}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (-\frac {-1+2 b}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (\operatorname {BesselY}\left (\frac {\operatorname {csgn}\left (2 b -1\right ) \left (2 b -1\right )}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) c_{2} +\operatorname {BesselJ}\left (\frac {\operatorname {csgn}\left (2 b -1\right ) \left (2 b -1\right )}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) c_{1} \right ) \]

Problem 9485

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\frac {y}{\ln \left (x \right )}=x \,{\mathrm e}^{x} \left (2+x \ln \left (x \right )\right )} \]
program solution
Maple solution	\[ y \left (x \right ) = -\ln \left (x \right )^{3} \operatorname {expIntegral}_{1}\left (-\ln \left (x \right )\right ) {\mathrm e}^{x}-\ln \left (x \right )^{2} {\mathrm e}^{x} x -\ln \left (x \right ) \operatorname {expIntegral}_{1}\left (-\ln \left (x \right )\right ) c_{1} +\left (\int \frac {\left (\operatorname {expIntegral}_{1}\left (-\ln \left (x \right )\right ) \ln \left (x \right )+x \right ) {\mathrm e}^{x} \left (2+x \ln \left (x \right )\right )}{x}d x \right ) \ln \left (x \right )+\ln \left (x \right ) c_{2} -c_{1} x \]

Problem 9486

ODE	\[ \boxed {x^{2} y^{\prime \prime }+a y^{\prime }-x y=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 9487

ODE	\[ \boxed {x^{2} y^{\prime \prime }+a y^{\prime }-\left (b^{2} x^{2}+a b \right ) y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{b x}+c_{2} {\mathrm e}^{b x} \left (\int {\mathrm e}^{\frac {-2 b \,x^{2}+a}{x}}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (\operatorname {HeunD}\left (4 \sqrt {2}\, \sqrt {a b}, -1-4 \sqrt {2}\, \sqrt {a b}, 8 \sqrt {2}\, \sqrt {a b}, -4 \sqrt {2}\, \sqrt {a b}+1, \frac {\sqrt {2}\, \sqrt {a b}\, x -a}{\sqrt {2}\, \sqrt {a b}\, x +a}\right ) {\mathrm e}^{\frac {-b \,x^{2}+a}{x}} c_{1} +{\mathrm e}^{b x} \operatorname {HeunD}\left (-4 \sqrt {2}\, \sqrt {a b}, -1-4 \sqrt {2}\, \sqrt {a b}, 8 \sqrt {2}\, \sqrt {a b}, -4 \sqrt {2}\, \sqrt {a b}+1, \frac {\sqrt {2}\, \sqrt {a b}\, x -a}{\sqrt {2}\, \sqrt {a b}\, x +a}\right ) c_{2} \right ) \]

Problem 9488

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x -y=x^{2} a} \]
program solution	\[ y = x \left (\frac {x a}{3}-\frac {c_{1}}{2 x^{2}}\right )+c_{2} x \] Verified OK.
Maple solution	\[ y \left (x \right ) = x c_{2} +\frac {a \,x^{2}}{3}+\frac {c_{1}}{x} \]

Problem 9489

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +a y=0} \]
program solution	\[ y = c_{1} x^{\sqrt {-a}}-\frac {c_{2} x^{-\sqrt {-a}}}{2 \sqrt {-a}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sin \left (\ln \left (x \right ) \sqrt {a}\right )+c_{2} \cos \left (\ln \left (x \right ) \sqrt {a}\right ) \]

Problem 9490

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x -\left (x +a \right ) y=0} \]
program solution	\[ y = c_{1} \operatorname {BesselJ}\left (2 \sqrt {a}, 2 i \sqrt {x}\right )+c_{2} \operatorname {BesselY}\left (2 \sqrt {a}, 2 i \sqrt {x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {BesselI}\left (2 \sqrt {a}, 2 \sqrt {x}\right )+c_{2} \operatorname {BesselK}\left (2 \sqrt {a}, 2 \sqrt {x}\right ) \]

Problem 9491

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (-v^{2}+x^{2}\right ) y=0} \]
program solution	\[ y = c_{1} \operatorname {BesselJ}\left (v , x\right )+c_{2} \operatorname {BesselY}\left (v , x\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {BesselJ}\left (v , x\right )+c_{2} \operatorname {BesselY}\left (v , x\right ) \]

Problem 9492

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (-v^{2}+x^{2}\right ) y=f \left (x \right )} \]
program solution	\[ y = c_{1} \operatorname {BesselJ}\left (v , x\right )+c_{2} \operatorname {BesselY}\left (v , x\right )-\frac {\pi \left (\left (\int _{0}^{x}\frac {\operatorname {BesselY}\left (v , \alpha \right ) f \left (\alpha \right )}{\alpha }d \alpha \right ) \operatorname {BesselJ}\left (v , x\right )-\left (\int _{0}^{x}\frac {\operatorname {BesselJ}\left (v , \alpha \right ) f \left (\alpha \right )}{\alpha }d \alpha \right ) \operatorname {BesselY}\left (v , x\right )\right )}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {BesselJ}\left (v , x\right ) c_{2} +\operatorname {BesselY}\left (v , x\right ) c_{1} +\frac {\pi \left (\int \frac {\operatorname {BesselJ}\left (v , x\right ) f \left (x \right )}{x}d x \right ) \operatorname {BesselY}\left (v , x\right )}{2}-\frac {\pi \left (\int \frac {\operatorname {BesselY}\left (v , x\right ) f \left (x \right )}{x}d x \right ) \operatorname {BesselJ}\left (v , x\right )}{2} \]

Problem 9493

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (l \,x^{2}-v^{2}\right ) y=0} \]
program solution	\[ y = c_{1} \operatorname {BesselJ}\left (v , x \sqrt {l}\right )+c_{2} \operatorname {BesselY}\left (v , x \sqrt {l}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {BesselJ}\left (v , \sqrt {l}\, x \right )+c_{2} \operatorname {BesselY}\left (v , \sqrt {l}\, x \right ) \]

Problem 9494

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (x +a \right ) y^{\prime }-y=0} \]
program solution	\[ y = \frac {c_{2} x \,{\mathrm e}^{\frac {a}{x}} a^{2}+c_{1} \left (x +a \right )}{a^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (a +x \right ) c_{1} +c_{2} x \,{\mathrm e}^{\frac {a}{x}} \]

Problem 9495

ODE	\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x +y=3 x^{3}} \]
program solution	\[ y = x \left (c_{1} +c_{2} \ln \left (x \right )\right )+\frac {3 x^{3}}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x \left (4 c_{1} \ln \left (x \right )+3 x^{2}+4 c_{2} \right )}{4} \]

Problem 9496

ODE	\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x +\left (a \,x^{m}+b \right ) y=0} \]
program solution	\[ y = c_{1} x \operatorname {BesselJ}\left (\frac {2 \sqrt {-b +1}}{m}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}}}{m}\right )+c_{2} x \operatorname {BesselY}\left (\frac {2 \sqrt {-b +1}}{m}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}}}{m}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x \left (c_{1} \operatorname {BesselJ}\left (\frac {2 \sqrt {1-b}}{m}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}}}{m}\right )+c_{2} \operatorname {BesselY}\left (\frac {2 \sqrt {1-b}}{m}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}}}{m}\right )\right ) \]

Problem 9497

ODE	\[ \boxed {x^{2} y^{\prime \prime }+2 y^{\prime } x=0} \]
program solution	\[ y = -\frac {c_{1}}{x}+c_{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} +\frac {c_{2}}{x} \]

Problem 9498

ODE	\[ \boxed {x^{2} y^{\prime \prime }+2 y^{\prime } x +\left (x a -b^{2}\right ) y=0} \]
program solution	\[ y = \frac {c_{1} \operatorname {BesselJ}\left (\sqrt {4 b^{2}+1}, 2 \sqrt {a}\, \sqrt {x}\right )}{\sqrt {x}}+\frac {c_{2} \operatorname {BesselY}\left (\sqrt {4 b^{2}+1}, 2 \sqrt {a}\, \sqrt {x}\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \operatorname {BesselJ}\left (\sqrt {4 b^{2}+1}, 2 \sqrt {a}\, \sqrt {x}\right )+c_{2} \operatorname {BesselY}\left (\sqrt {4 b^{2}+1}, 2 \sqrt {a}\, \sqrt {x}\right )}{\sqrt {x}} \]

Problem 9499

ODE	\[ \boxed {x^{2} y^{\prime \prime }+2 y^{\prime } x +\left (x^{2} a +b \right ) y=0} \]
program solution	\[ y = \frac {c_{1} \operatorname {BesselJ}\left (\frac {\sqrt {-4 b +1}}{2}, x \sqrt {a}\right )}{\sqrt {x}}+\frac {c_{2} \operatorname {BesselY}\left (\frac {\sqrt {-4 b +1}}{2}, x \sqrt {a}\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \operatorname {BesselJ}\left (\frac {\sqrt {1-4 b}}{2}, x \sqrt {a}\right )+c_{2} \operatorname {BesselY}\left (\frac {\sqrt {1-4 b}}{2}, x \sqrt {a}\right )}{\sqrt {x}} \]

Problem 9500

ODE	\[ \boxed {x^{2} y^{\prime \prime }+2 y^{\prime } x +\left (l \,x^{2}+x a -n \left (1+n \right )\right ) y=0} \]
program solution	\[ y = \frac {c_{1} \operatorname {BesselJ}\left (-1-2 n , 2 \sqrt {x}\right )}{\sqrt {x}}+\frac {c_{2} \operatorname {BesselY}\left (-1-2 n , 2 \sqrt {x}\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} \operatorname {WhittakerW}\left (-\frac {i a}{2 \sqrt {l}}, n +\frac {1}{2}, 2 i \sqrt {l}\, x \right )+c_{1} \operatorname {WhittakerM}\left (-\frac {i a}{2 \sqrt {l}}, n +\frac {1}{2}, 2 i \sqrt {l}\, x \right )}{x} \]