2.17.23 Problems 2201 to 2300

Problem 2201

ODE	\[ \boxed {y^{\prime \prime }-5 y^{\prime }+6 y=x^{2}} \]
program solution	\[ y = c_{1} {\mathrm e}^{2 x}+c_{2} {\mathrm e}^{3 x}+\frac {x^{2}}{6}+\frac {5 x}{18}+\frac {19}{108} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} {\mathrm e}^{3 x}+c_{1} {\mathrm e}^{2 x}+\frac {x^{2}}{6}+\frac {5 x}{18}+\frac {19}{108} \]

Problem 2202

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

Problem 2203

ODE	\[ \boxed {y^{\prime \prime }+3 y=3 \,{\mathrm e}^{-4 x}} \]
program solution	\[ y = c_{1} \cos \left (\sqrt {3}\, x \right )+\frac {c_{2} \sin \left (\sqrt {3}\, x \right ) \sqrt {3}}{3}+\frac {3 \,{\mathrm e}^{-4 x}}{19} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sin \left (\sqrt {3}\, x \right ) c_{2} +\cos \left (\sqrt {3}\, x \right ) c_{1} +\frac {3 \,{\mathrm e}^{-4 x}}{19} \]

Problem 2204

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+4 y=\frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{-x}}{2}} \]
program solution	\[ y = {\mathrm e}^{-2 x} \left (c_{2} x +c_{1} \right )+\frac {{\mathrm e}^{x}}{18}+\frac {{\mathrm e}^{-x}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left ({\mathrm e}^{3 x}+18 c_{1} x +9 \,{\mathrm e}^{x}+18 c_{2} \right ) {\mathrm e}^{-2 x}}{18} \]

Problem 2205

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

Problem 2206

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

Problem 2207

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+4 y=\frac {{\mathrm e}^{3 x}}{2}-\frac {{\mathrm e}^{-3 x}}{2}} \]
program solution	\[ y = {\mathrm e}^{-2 x} \left (c_{2} x +c_{1} \right )-\frac {{\mathrm e}^{-3 x}}{2}+\frac {{\mathrm e}^{3 x}}{50} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-25+{\mathrm e}^{6 x}+50 \,{\mathrm e}^{x} \left (c_{1} x +c_{2} \right )\right ) {\mathrm e}^{-3 x}}{50} \]

Problem 2208

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }-2 y=\sin \left (2 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {\left (3+\sqrt {17}\right ) x}{2}}+\frac {c_{2} \sqrt {17}\, {\mathrm e}^{\frac {\left (-3+\sqrt {17}\right ) x}{2}}}{17}-\frac {\cos \left (2 x \right )}{12}-\frac {\sin \left (2 x \right )}{12} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{\frac {\left (-3+\sqrt {17}\right ) x}{2}} c_{2} +{\mathrm e}^{-\frac {\left (3+\sqrt {17}\right ) x}{2}} c_{1} -\frac {\sin \left (2 x \right )}{12}-\frac {\cos \left (2 x \right )}{12} \]

Problem 2209

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=\sin \left (x \right ) {\mathrm e}^{x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{-x}-\frac {\cos \left (x \right ) {\mathrm e}^{x}}{10}+\frac {\sin \left (x \right ) {\mathrm e}^{x}}{10} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -{\mathrm e}^{-2 x} \left (\frac {\left (-\sin \left (x \right )+\cos \left (x \right )\right ) {\mathrm e}^{3 x}}{10}-c_{2} {\mathrm e}^{x}+c_{1} \right ) \]

Problem 2210

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

Problem 2211

ODE	\[ \boxed {y^{\prime \prime \prime }-4 y^{\prime \prime }+y^{\prime }-4 y=\sin \left (x \right )-{\mathrm e}^{4 x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{4 x}+{\mathrm e}^{-i x} c_{2} +{\mathrm e}^{i x} c_{3} +\frac {\left (-17 x +8\right ) {\mathrm e}^{4 x}}{289}+\frac {\left (9+68 i+272 x \right ) \cos \left (x \right )}{2312}-\frac {\sin \left (x \right ) \left (\frac {236}{17}+i+4 x \right )}{136} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (8-17 x +289 c_{3} \right ) {\mathrm e}^{4 x}}{289}+\frac {\left (68 x +578 c_{1} +15\right ) \cos \left (x \right )}{578}-\frac {\sin \left (x \right ) \left (x -34 c_{2} +\frac {8}{17}\right )}{34} \]

Problem 2212

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+3 y^{\prime \prime }-4 y=4 \,{\mathrm e}^{x}+3 \cos \left (2 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+c_{2} {\mathrm e}^{x}+{\mathrm e}^{-2 i x} c_{3} +{\mathrm e}^{2 i x} c_{4} -\frac {63 \cos \left (2 x \right )}{400}-\frac {3 x \sin \left (2 x \right )}{20}+\frac {\left (10 x -9\right ) {\mathrm e}^{x}}{25} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-3+25 c_{2} \right ) \cos \left (2 x \right )}{25}+\frac {\left (-3 x +20 c_{4} \right ) \sin \left (2 x \right )}{20}+c_{3} {\mathrm e}^{-x}+\frac {2 \,{\mathrm e}^{x} \left (x +\frac {5 c_{1}}{2}-\frac {9}{10}\right )}{5} \]

Problem 2213

ODE	\[ \boxed {y^{\prime \prime }+y={\mathrm e}^{3 x} \left (1+\sin \left (2 x \right )\right )} \]
program solution	\[ y = c_{1} \cos \left (x \right )+c_{2} \sin \left (x \right )+\frac {{\mathrm e}^{3 x}}{10}-\frac {{\mathrm e}^{3 x} \cos \left (2 x \right )}{15}+\frac {{\mathrm e}^{3 x} \sin \left (2 x \right )}{30} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (2 \cos \left (x \right ) \sin \left (x \right )+5-4 \cos \left (x \right )^{2}\right ) {\mathrm e}^{3 x}}{30}+\cos \left (x \right ) c_{1} +\sin \left (x \right ) c_{2} \]

Problem 2214

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

Problem 2215

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+5 y=\frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{-x}}{2}} \]
program solution	\[ y = {\mathrm e}^{-2 x} \left (c_{1} \cos \left (x \right )+c_{2} \sin \left (x \right )\right )+\frac {{\mathrm e}^{x}}{20}+\frac {{\mathrm e}^{-x}}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left ({\mathrm e}^{3 x}+20 \sin \left (x \right ) c_{2} +20 \cos \left (x \right ) c_{1} +5 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-2 x}}{20} \]

Problem 2216

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

Problem 2217

ODE	\[ \boxed {y^{\prime \prime }+4 y=x \,{\mathrm e}^{x}} \]
program solution	\[ y = c_{1} \cos \left (2 x \right )+\frac {c_{2} \sin \left (2 x \right )}{2}+\frac {x \,{\mathrm e}^{x}}{5}-\frac {2 \,{\mathrm e}^{x}}{25} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sin \left (2 x \right ) c_{2} +c_{1} \cos \left (2 x \right )+\frac {\left (5 x -2\right ) {\mathrm e}^{x}}{25} \]

Problem 2218

ODE	\[ \boxed {y^{\prime \prime }+2 y=x^{2} {\mathrm e}^{-x}} \]
program solution	\[ y = c_{1} \cos \left (\sqrt {2}\, x \right )+\frac {c_{2} \sqrt {2}\, \sin \left (\sqrt {2}\, x \right )}{2}+\frac {4 x \,{\mathrm e}^{-x}}{9}+\frac {x^{2} {\mathrm e}^{-x}}{3}+\frac {2 \,{\mathrm e}^{-x}}{27} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \cos \left (\sqrt {2}\, x \right ) c_{1} +c_{2} \sin \left (\sqrt {2}\, x \right )+\frac {\left (x^{2}+\frac {4}{3} x +\frac {2}{9}\right ) {\mathrm e}^{-x}}{3} \]

Problem 2219

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }-2 y=x^{2}-8} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+\frac {c_{2} {\mathrm e}^{2 x}}{3}-\frac {x^{2}}{2}+\frac {x}{2}+\frac {13}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} {\mathrm e}^{2 x}+{\mathrm e}^{-x} c_{1} -\frac {x^{2}}{2}+\frac {x}{2}+\frac {13}{4} \]

Problem 2220

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

Problem 2221

ODE	\[ \boxed {y^{\prime \prime \prime }+4 y^{\prime \prime }-5 y^{\prime }=x^{2} {\mathrm e}^{-x}} \]
program solution	\[ y = c_{2} {\mathrm e}^{x}+c_{1} +{\mathrm e}^{-5 x} c_{3} +\frac {5 x \,{\mathrm e}^{-x}}{16}+\frac {x^{2} {\mathrm e}^{-x}}{8}+\frac {23 \,{\mathrm e}^{-x}}{64} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\left (x^{2}+\frac {5}{2} x +\frac {23}{8}\right ) {\mathrm e}^{4 x}+8 \,{\mathrm e}^{6 x} c_{2} +8 c_{3} {\mathrm e}^{5 x}-\frac {8 c_{1}}{5}\right ) {\mathrm e}^{-5 x}}{8} \]

Problem 2222

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

Problem 2223

ODE	\[ \boxed {y^{\prime \prime \prime }-y^{\prime }={\mathrm e}^{x} \left (\sin \left (x \right )-x^{2}\right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+c_{2} +{\mathrm e}^{x} c_{3} -\frac {\cos \left (x \right ) {\mathrm e}^{x}}{10}-\frac {3 \sin \left (x \right ) {\mathrm e}^{x}}{10}-\frac {7 x \,{\mathrm e}^{x}}{4}+\frac {3 x^{2} {\mathrm e}^{x}}{4}-\frac {{\mathrm e}^{x} x^{3}}{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -{\mathrm e}^{-x} c_{1} +\frac {\left (-20 x^{3}+90 x^{2}-210 x +120 c_{2} -12 \cos \left (x \right )-36 \sin \left (x \right )+225\right ) {\mathrm e}^{x}}{120}+c_{3} \]

Problem 2224

ODE	\[ \boxed {y^{\prime \prime \prime }-4 y^{\prime \prime }={\mathrm e}^{2 x} \left (x -3\right )} \]
program solution	\[ y = c_{2} x +c_{1} +{\mathrm e}^{4 x} c_{3} -\frac {x \,{\mathrm e}^{2 x}}{8}+\frac {7 \,{\mathrm e}^{2 x}}{16} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-2 x +7\right ) {\mathrm e}^{2 x}}{16}+c_{2} x +\frac {{\mathrm e}^{4 x} c_{1}}{16}+c_{3} \]

Problem 2225

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-6 y^{\prime \prime \prime }+9 y^{\prime \prime }=\sin \left (3 x \right )+x \,{\mathrm e}^{x}} \]
program solution	\[ y = \left (c_{4} x +c_{3} \right ) {\mathrm e}^{3 x}+c_{2} x +c_{1} +\frac {x \,{\mathrm e}^{x}}{4}-\frac {{\mathrm e}^{x}}{4}-\frac {\cos \left (3 x \right )}{162} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (3 c_{1} x -2 c_{1} +3 c_{2} \right ) {\mathrm e}^{3 x}}{27}-\frac {\cos \left (3 x \right )}{162}+\frac {\left (x -1\right ) {\mathrm e}^{x}}{4}+c_{3} x +c_{4} \]

Problem 2226

ODE	\[ \boxed {y^{\prime \prime \prime }-6 y^{\prime \prime }+11 y^{\prime }-6 y=x^{2} {\mathrm e}^{2 x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{2 x}+{\mathrm e}^{3 x} c_{3} -2 x \,{\mathrm e}^{2 x}-\frac {{\mathrm e}^{2 x} x^{3}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-x^{3}+3 c_{2} -6 x \right ) {\mathrm e}^{2 x}}{3}+{\mathrm e}^{x} c_{1} +c_{3} {\mathrm e}^{3 x} \]

Problem 2227

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

Problem 2228

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+3 y^{\prime \prime }-y^{\prime }+2 y=\sin \left (2 x \right )} \]
program solution	\[ y = {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} =3\right ) x} c_{1} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} =1\right ) x} c_{2} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} =2\right ) x} c_{3} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} =4\right ) x} c_{4} +\frac {\cos \left (2 x \right )}{20}+\frac {3 \sin \left (2 x \right )}{20} \] Verified OK.
Maple solution	\[ \text {Expression too large to display} \]

Problem 2229

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+y^{\prime }=x^{3}-\frac {\cos \left (2 x \right )}{2}} \]
program solution	\[ y = c_{1} +{\mathrm e}^{\frac {\left (i \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}} \sqrt {3}+\left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}}+24 i \sqrt {3}-24\right ) x}{12 \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}}}} c_{2} +{\mathrm e}^{-\frac {\left (\left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}}-24\right ) x}{6 \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}}}} c_{3} +{\mathrm e}^{-\frac {x \left (\left (i \sqrt {3}-1\right ) \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}}+24 i \sqrt {3}+24\right )}{12 \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}}}} c_{4} -\frac {\cos \left (2 x \right )}{17}-\frac {\sin \left (2 x \right )}{68}-54 x +12 x^{2}-2 x^{3}+\frac {x^{4}}{4} \] Verified OK.
Maple solution	\[ \text {Expression too large to display} \]

Problem 2230

ODE	\[ \boxed {y^{\prime \prime \prime }+4 y^{\prime \prime }+5 y^{\prime }={\mathrm e}^{-2 x} \cos \left (x \right )} \]
program solution	\[ y = c_{1} +{\mathrm e}^{\left (-2-i\right ) x} c_{2} +{\mathrm e}^{\left (-2+i\right ) x} c_{3} +\frac {{\mathrm e}^{-2 x} \left (\left (-\frac {26}{5}+i-4 x \right ) \cos \left (x \right )+2 \sin \left (x \right ) \left (-\frac {7}{10}+i-4 x \right )\right )}{40} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\left (-5 x -20 c_{1} -10 c_{2} -4\right ) \cos \left (x \right )-10 \sin \left (x \right ) \left (x -c_{1} +2 c_{2} +\frac {3}{10}\right )\right ) {\mathrm e}^{-2 x}}{50}+c_{3} \]

Problem 2231

ODE	\[ \boxed {y^{\prime \prime \prime }+y^{\prime \prime }-2 y^{\prime }={\mathrm e}^{-2 x} \cos \left (2 x \right )} \]
program solution	\[ y = c_{1} +c_{2} {\mathrm e}^{-2 x}+{\mathrm e}^{x} c_{3} +\frac {5 \,{\mathrm e}^{-2 x} \cos \left (2 x \right )}{104}+\frac {{\mathrm e}^{-2 x} \sin \left (2 x \right )}{104} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {{\mathrm e}^{-2 x} \left (-2 c_{2} {\mathrm e}^{3 x}-2 c_{3} {\mathrm e}^{2 x}+c_{1} -\frac {5 \cos \left (2 x \right )}{52}-\frac {\sin \left (2 x \right )}{52}\right )}{2} \]

Problem 2232

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

Problem 2233

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-y=x^{2} \cos \left (x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+c_{2} {\mathrm e}^{x}+{\mathrm e}^{-i x} c_{3} +{\mathrm e}^{i x} c_{4} +\frac {\sin \left (x \right ) \left (-8 x^{3}+60 x +3 i\right )}{96}+\frac {\left (-36 x^{2}+33\right ) \cos \left (x \right )}{96} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{4} {\mathrm e}^{-x}+\frac {\left (-3 x^{2}+8 c_{1} +2\right ) \cos \left (x \right )}{8}+\frac {\left (-2 x^{3}+24 c_{3} +15 x \right ) \sin \left (x \right )}{24}+c_{2} {\mathrm e}^{x} \]

Problem 2234

ODE	\[ \boxed {y^{\prime \prime }+4 y=\sin \left (x \right ) x} \]
program solution	\[ y = c_{1} \cos \left (2 x \right )+\frac {c_{2} \sin \left (2 x \right )}{2}+\frac {\sin \left (x \right ) x}{3}-\frac {2 \cos \left (x \right )}{9} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sin \left (2 x \right ) c_{2} +c_{1} \cos \left (2 x \right )-\frac {2 \cos \left (x \right )}{9}+\frac {x \sin \left (x \right )}{3} \]

Problem 2235

ODE	\[ \boxed {y^{\prime \prime }+y=x^{2} \cos \left (x \right )} \]
program solution	\[ y = c_{1} \cos \left (x \right )+c_{2} \sin \left (x \right )+\frac {x^{2} \cos \left (x \right )}{4}+\frac {x^{3} \sin \left (x \right )}{6}-\frac {\sin \left (x \right ) x}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (2 x^{3}+12 c_{2} -3 x \right ) \sin \left (x \right )}{12}+\frac {\cos \left (x \right ) \left (x^{2}+4 c_{1} \right )}{4} \]

Problem 2236

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

Problem 2237

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

Problem 2238

ODE	\[ \boxed {y^{\left (5\right )}+y^{\prime \prime \prime \prime }=x^{2}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+c_{2} +c_{3} x +c_{4} x^{2}+c_{5} x^{3}+\frac {x^{6}}{360}-\frac {x^{5}}{60}+\frac {x^{4}}{12} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{6}}{360}+\frac {x^{4}}{12}-\frac {x^{5}}{60}+\frac {c_{2} x^{3}}{6}+\frac {c_{3} x^{2}}{2}+{\mathrm e}^{-x} c_{1} +c_{4} x +c_{5} \]

Problem 2239

ODE	\[ \boxed {2 y^{\prime \prime }+3 y^{\prime }-2 y=x^{2} {\mathrm e}^{x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+\frac {2 c_{2} {\mathrm e}^{\frac {x}{2}}}{5}-\frac {14 x \,{\mathrm e}^{x}}{9}+\frac {x^{2} {\mathrm e}^{x}}{3}+\frac {86 \,{\mathrm e}^{x}}{27} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{-2 x} \left (3 c_{2} {\mathrm e}^{\frac {5 x}{2}}+\left (x^{2}-\frac {14}{3} x +\frac {86}{9}\right ) {\mathrm e}^{3 x}+3 c_{1} \right )}{3} \]

Problem 2240

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

Problem 2241

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

Problem 2242

ODE	\[ \boxed {y^{\prime \prime \prime }+2 y^{\prime \prime }=x \cos \left (2 x \right )} \]
program solution	\[ y = c_{2} x +c_{1} +{\mathrm e}^{-2 x} c_{3} -\frac {x \cos \left (2 x \right )}{16}-\frac {x \sin \left (2 x \right )}{16}-\frac {\cos \left (2 x \right )}{16}+\frac {3 \sin \left (2 x \right )}{32} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-2 x -2\right ) \cos \left (2 x \right )}{32}+\frac {\left (3-2 x \right ) \sin \left (2 x \right )}{32}+c_{2} x +\frac {{\mathrm e}^{-2 x} c_{1}}{4}+c_{3} \]

Problem 2243

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=x^{2} \cos \left (x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{-x}+\frac {6 x \cos \left (x \right )}{25}+\frac {x^{2} \cos \left (x \right )}{10}-\frac {17 \sin \left (x \right ) x}{25}+\frac {3 \sin \left (x \right ) x^{2}}{10}-\frac {133 \cos \left (x \right )}{250}+\frac {81 \sin \left (x \right )}{250} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -{\mathrm e}^{-2 x} c_{1} +{\mathrm e}^{-x} c_{2} +\frac {\left (25 x^{2}+60 x -133\right ) \cos \left (x \right )}{250}+\frac {\sin \left (x \right ) \left (75 x^{2}-170 x +81\right )}{250} \]

Problem 2244

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+3 y=\sin \left (x \right ) x^{2}} \]
program solution	\[ y = c_{1} {\mathrm e}^{x}+\frac {c_{2} {\mathrm e}^{3 x}}{2}+\frac {11 x \cos \left (x \right )}{25}+\frac {x^{2} \cos \left (x \right )}{5}-\frac {2 \sin \left (x \right ) x}{25}+\frac {\sin \left (x \right ) x^{2}}{10}+\frac {28 \cos \left (x \right )}{125}-\frac {67 \sin \left (x \right )}{250} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} {\mathrm e}^{3 x}+\frac {\left (25 x^{2}+55 x +28\right ) \cos \left (x \right )}{125}+\frac {\left (25 x^{2}-20 x -67\right ) \sin \left (x \right )}{250}+{\mathrm e}^{x} c_{1} \]

Problem 2245

ODE	\[ \boxed {y^{\prime \prime }-y=x \sin \left (2 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+\frac {c_{2} {\mathrm e}^{x}}{2}-\frac {x \sin \left (2 x \right )}{5}-\frac {4 \cos \left (2 x \right )}{25} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-x} c_{2} +{\mathrm e}^{x} c_{1} -\frac {4 \cos \left (2 x \right )}{25}-\frac {\sin \left (2 x \right ) x}{5} \]

Problem 2246

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }=x^{3} \sin \left (2 x \right )} \]
program solution	\[ y = \frac {\left (-8 x^{3}+24 x^{2}+6 x -15\right ) \sin \left (2 x \right )}{64}+\frac {\left (-4 x^{3}-6 x^{2}+15 x \right ) \cos \left (2 x \right )}{32}+c_{2} {\mathrm e}^{-2 x}+\frac {c_{1}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-8 x^{3}+24 x^{2}+6 x -15\right ) \sin \left (2 x \right )}{64}+\frac {\left (-4 x^{3}-6 x^{2}+15 x \right ) \cos \left (2 x \right )}{32}-\frac {{\mathrm e}^{-2 x} c_{1}}{2}+c_{2} \]

Problem 2247

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }={\mathrm e}^{2 x} \sin \left (x \right ) x} \]
program solution	\[ y = \frac {\left (\left (-15 x +17\right ) \cos \left (x \right )+\sin \left (x \right ) \left (5 x +6\right )\right ) {\mathrm e}^{2 x}}{50}+c_{2} {\mathrm e}^{x}-c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\left (-15 x +17\right ) \cos \left (x \right )+\sin \left (x \right ) \left (5 x +6\right )\right ) {\mathrm e}^{2 x}}{50}+{\mathrm e}^{x} c_{1} +c_{2} \]

Problem 2248

ODE	\[ \boxed {y^{\prime \prime }-4 y={\mathrm e}^{2 x} \cos \left (x \right ) x} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+\frac {c_{2} {\mathrm e}^{2 x}}{4}+\frac {76 \,{\mathrm e}^{2 x} \cos \left (x \right )}{289}+\frac {2 \,{\mathrm e}^{2 x} \sin \left (x \right )}{289}-\frac {{\mathrm e}^{2 x} \cos \left (x \right ) x}{17}+\frac {4 \,{\mathrm e}^{2 x} \sin \left (x \right ) x}{17} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\left (-17 x +76\right ) \cos \left (x \right )+\left (68 x +2\right ) \sin \left (x \right )+289 c_{2} \right ) {\mathrm e}^{2 x}}{289}+{\mathrm e}^{-2 x} c_{1} \]

Problem 2249

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

Problem 2250

ODE	\[ \boxed {x^{2} y^{\prime \prime }-4 y^{\prime } x +y=0} \]
program solution	\[ y = c_{1} x^{\frac {5}{2}-\frac {\sqrt {21}}{2}}+\frac {c_{2} \sqrt {21}\, x^{\frac {5}{2}+\frac {\sqrt {21}}{2}}}{21} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{\frac {5}{2}} \left (x^{\frac {\sqrt {21}}{2}} c_{1} +x^{-\frac {\sqrt {21}}{2}} c_{2} \right ) \]

Problem 2251

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +16 y=0} \]
program solution	\[ y = x^{-4 i} c_{1} -\frac {i c_{2} x^{4 i}}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sin \left (4 \ln \left (x \right )\right )+c_{2} \cos \left (4 \ln \left (x \right )\right ) \]

Problem 2252

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }-16 y^{\prime } x +25 y=0} \]
program solution	\[ y = c_{1} x^{\frac {5}{2}}+c_{2} x^{\frac {5}{2}} \ln \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) x^{\frac {5}{2}} \]

Problem 2253

ODE	\[ \boxed {x^{2} y^{\prime \prime }+5 y^{\prime } x +10 y=0} \]
program solution	\[ y = c_{1} x^{-2-i \sqrt {6}}-\frac {i c_{2} \sqrt {6}\, x^{-2+i \sqrt {6}}}{12} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sin \left (\sqrt {6}\, \ln \left (x \right )\right )+c_{2} \cos \left (\sqrt {6}\, \ln \left (x \right )\right )}{x^{2}} \]

Problem 2254

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }-3 y^{\prime } x -18 y=\ln \left (x \right )} \]
program solution	\[ y = \frac {c_{1}}{x^{2}}+\frac {2 c_{2} x^{\frac {9}{2}}}{13}-\frac {\ln \left (x \right )}{18}+\frac {5}{324} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2}}{x^{2}}+x^{\frac {9}{2}} c_{1} -\frac {\ln \left (x \right )}{18}+\frac {5}{324} \]

Problem 2255

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

Problem 2256

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

Problem 2257

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

Problem 2258

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

Problem 2259

ODE	\[ \boxed {x^{2} y^{\prime \prime }-2 y^{\prime } x +2 y=4 x +\sin \left (\ln \left (x \right )\right )} \]
program solution	\[ y = x \left (c_{2} x +c_{1} \right )+\frac {\sin \left (\ln \left (x \right )\right )}{10}+\frac {3 \cos \left (\ln \left (x \right )\right )}{10}-4 x \ln \left (x \right )-4 x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {3 \cos \left (\ln \left (x \right )\right )}{10}+\frac {\sin \left (\ln \left (x \right )\right )}{10}-4 x \ln \left (x \right )+c_{2} x^{2}+\left (-4+c_{1} \right ) x \]

Problem 2260

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

Problem 2261

ODE	\[ \boxed {x^{2} y^{\prime \prime }+4 y^{\prime } x +3 y=\left (x -1\right ) \ln \left (x \right )} \]
program solution	\[ y = c_{1} x^{-\frac {i \sqrt {3}}{2}-\frac {3}{2}}-\frac {i c_{2} \sqrt {3}\, x^{\frac {i \sqrt {3}}{2}-\frac {3}{2}}}{3}+\frac {1}{3}+\frac {\left (3 x -7\right ) \ln \left (x \right )}{21}-\frac {5 x}{49} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\sin \left (\frac {\sqrt {3}\, \ln \left (x \right )}{2}\right ) c_{2}}{x^{\frac {3}{2}}}+\frac {\cos \left (\frac {\sqrt {3}\, \ln \left (x \right )}{2}\right ) c_{1}}{x^{\frac {3}{2}}}+\frac {1}{3}+\frac {\left (3 x -7\right ) \ln \left (x \right )}{21}-\frac {5 x}{49} \]

Problem 2262

ODE	\[ \boxed {4 x^{3} y^{\prime \prime \prime }+8 x^{2} y^{\prime \prime }-y^{\prime } x +y=x +\ln \left (x \right )} \]
program solution	\[ y = c_{1} x +\frac {c_{2}}{\sqrt {x}}+c_{3} \sqrt {x}+\frac {x \ln \left (x \right )}{3}+\ln \left (x \right )+1-\frac {8 x}{9} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x \ln \left (x \right )}{3}+\ln \left (x \right )+1-\frac {8 x}{9}+c_{1} x +\frac {c_{2}}{\sqrt {x}}+c_{3} \sqrt {x} \]

Problem 2263

ODE	\[ \boxed {3 x^{3} y^{\prime \prime \prime }+4 x^{2} y^{\prime \prime }-10 y^{\prime } x +10 y=\frac {4}{x^{2}}} \]
program solution	\[ y = c_{1} x +c_{2} x^{\frac {1}{3}+\frac {\sqrt {31}}{3}}+c_{3} x^{\frac {1}{3}-\frac {\sqrt {31}}{3}}-\frac {2}{9 x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {9 c_{3} x^{\frac {7}{3}+\frac {\sqrt {31}}{3}}+9 c_{2} x^{\frac {7}{3}-\frac {\sqrt {31}}{3}}+9 c_{1} x^{3}-2}{9 x^{2}} \]

Problem 2264

ODE	\[ \boxed {x^{4} y^{\prime \prime \prime \prime }+7 x^{3} y^{\prime \prime \prime }+9 x^{2} y^{\prime \prime }-6 y^{\prime } x -6 y=\cos \left (\ln \left (x \right )\right )} \]
program solution	\[ y = \frac {c_{2} x^{3}+c_{3} \cos \left (\sqrt {2}\, \ln \left (x \right )\right )+c_{4} \sin \left (\sqrt {2}\, \ln \left (x \right )\right )+c_{1}}{x}+\frac {5 \sqrt {2}\, \left (\int \cos \left (\ln \left (x \right )\right ) \left (3 \sqrt {2}\, \cos \left (\sqrt {2}\, \ln \left (x \right )\right )+2 \sin \left (\sqrt {2}\, \ln \left (x \right )\right )\right )d x \right ) \cos \left (\sqrt {2}\, \ln \left (x \right )\right )+5 \sqrt {2}\, \left (\int \cos \left (\ln \left (x \right )\right ) \left (3 \sqrt {2}\, \sin \left (\sqrt {2}\, \ln \left (x \right )\right )-2 \cos \left (\sqrt {2}\, \ln \left (x \right )\right )\right )d x \right ) \sin \left (\sqrt {2}\, \ln \left (x \right )\right )-21 x \left (\cos \left (\ln \left (x \right )\right )+\frac {17 \sin \left (\ln \left (x \right )\right )}{21}\right )}{220 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\left (\left (-66-132 i+\left (-33+99 i\right ) \sqrt {2}\right ) x^{1-i}+\left (-66+132 i+\left (33+99 i\right ) \sqrt {2}\right ) x^{1+i}-360 i \sqrt {2}\, c_{2} x^{3}+240 c_{2} x^{3}-440 c_{1} \right ) x^{-i \sqrt {2}}+\left (\left (-66-132 i+\left (33-99 i\right ) \sqrt {2}\right ) x^{1-i}+\left (-66+132 i+\left (-33-99 i\right ) \sqrt {2}\right ) x^{1+i}+360 i \sqrt {2}\, c_{2} x^{3}+240 c_{2} x^{3}-440 c_{1} \right ) x^{i \sqrt {2}}+5280 c_{3} \right ) \cos \left (\sqrt {2}\, \ln \left (x \right )\right )+240 \sin \left (\sqrt {2}\, \ln \left (x \right )\right ) \left (\left (\left (\frac {11}{20}-\frac {11 i}{40}+\left (-\frac {33}{80}-\frac {11 i}{80}\right ) \sqrt {2}\right ) x^{1-i}+\left (-\frac {11}{20}-\frac {11 i}{40}+\left (-\frac {33}{80}+\frac {11 i}{80}\right ) \sqrt {2}\right ) x^{1+i}+i x^{3} c_{2} +\frac {3 \sqrt {2}\, c_{2} x^{3}}{2}-\frac {11 i c_{1}}{6}\right ) x^{-i \sqrt {2}}+\left (\left (-\frac {11}{20}+\frac {11 i}{40}+\left (-\frac {33}{80}-\frac {11 i}{80}\right ) \sqrt {2}\right ) x^{1-i}+\left (\frac {11}{20}+\frac {11 i}{40}+\left (-\frac {33}{80}+\frac {11 i}{80}\right ) \sqrt {2}\right ) x^{1+i}-i x^{3} c_{2} +\frac {3 \sqrt {2}\, c_{2} x^{3}}{2}+\frac {11 i c_{1}}{6}\right ) x^{i \sqrt {2}}+22 c_{4} \right )}{5280 x} \]

Problem 2265

ODE	\[ \boxed {x^{3} y^{\prime \prime \prime }-2 x^{2} y^{\prime \prime }-y^{\prime } x +4 y=\sin \left (\ln \left (x \right )\right )} \]
program solution	\[ y = c_{1} x^{4}+c_{2} x^{\frac {1}{2}+\frac {\sqrt {5}}{2}}+c_{3} x^{\frac {1}{2}-\frac {\sqrt {5}}{2}}-\frac {9 x^{-i} \left (\left (\frac {2}{9}+i\right ) x^{2 i}+\frac {2}{9}-i\right )}{170} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} x^{\frac {1}{2}-\frac {\sqrt {5}}{2}}+c_{3} x^{\frac {1}{2}+\frac {\sqrt {5}}{2}}+\left (-\frac {1}{85}+\frac {9 i}{170}\right ) x^{-i}+\left (-\frac {1}{85}-\frac {9 i}{170}\right ) x^{i}+c_{1} x^{4} \]

Problem 2266

ODE	\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )+\cos \left (t \right )\\ y^{\prime }\left (t \right )&=-y \left (t \right )+4 t \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= -\frac {\cos \left (t \right )}{2}+\frac {\sin \left (t \right )}{2}+c_{1} {\mathrm e}^{t} \\ y \left (t \right ) &= 4 t -4+c_{2} {\mathrm e}^{-t} \\ \end{align*}

Problem 2267

ODE	\begin {align*} x^{\prime }\left (t \right )&=3 t^{2}-5 x \left (t \right )\\ y^{\prime }\left (t \right )&=-y \left (t \right )+{\mathrm e}^{3 t} \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= \frac {3 t^{2}}{5}-\frac {6 t}{25}+\frac {6}{125}+c_{2} {\mathrm e}^{-5 t} \\ y \left (t \right ) &= \frac {{\mathrm e}^{3 t}}{4}+{\mathrm e}^{-t} c_{1} \\ \end{align*}

Problem 2268

ODE	\begin {align*} x^{\prime }\left (t \right )&=-2 x \left (t \right )+3 t\\ y^{\prime }\left (t \right )&=x \left (t \right )-\frac {3 t}{2}-\frac {y \left (t \right )}{2}+\cos \left (t \right )^{2}-\frac {1}{2} \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= \frac {3 t}{2}-\frac {3}{4}+c_{2} {\mathrm e}^{-2 t} \\ y \left (t \right ) &= \frac {\cos \left (2 t \right )}{17}+\frac {4 \sin \left (2 t \right )}{17}-\frac {2 c_{2} {\mathrm e}^{-2 t}}{3}-\frac {3}{2}+c_{1} {\mathrm e}^{-\frac {t}{2}} \\ \end{align*}

Problem 2269

ODE	\begin {align*} x^{\prime }\left (t \right )&=-y \left (t \right )+x \left (t \right )+2 \sin \left (t \right )\\ y^{\prime }\left (t \right )&=4 y \left (t \right )-4 x \left (t \right )-2 \sin \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= -\frac {{\mathrm e}^{5 t} c_{1}}{20}-\frac {16 \cos \left (t \right )}{13}-\frac {2 \sin \left (t \right )}{13}+c_{2} \\ y \left (t \right ) &= \frac {{\mathrm e}^{5 t} c_{1}}{5}+\frac {8 \sin \left (t \right )}{13}-\frac {14 \cos \left (t \right )}{13}+c_{2} \\ \end{align*}

Problem 2270

ODE	\begin {align*} x^{\prime }\left (t \right )&=-\frac {3 x \left (t \right )}{2}+\frac {y \left (t \right )}{2}+\frac {{\mathrm e}^{t}}{2}\\ y^{\prime }\left (t \right )&=\frac {5 x \left (t \right )}{3}-\frac {y \left (t \right )}{3}-\frac {2 t}{3} \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= -c_{2} {\mathrm e}^{-2 t}+\frac {3 \,{\mathrm e}^{\frac {t}{6}} c_{1}}{10}+\frac {4 \,{\mathrm e}^{t}}{15}+\frac {11}{2}+t \\ y \left (t \right ) &= c_{2} {\mathrm e}^{-2 t}+{\mathrm e}^{\frac {t}{6}} c_{1} +\frac {37}{2}+\frac {{\mathrm e}^{t}}{3}+3 t \\ \end{align*}

Problem 2271

ODE	\begin {align*} 0&=-5 y^{\prime }\left (t \right )+3 x^{\prime }\left (t \right )+5 y \left (t \right )+5 t\\ 0&=-3 x^{\prime }\left (t \right )+5 y^{\prime }\left (t \right )+2 x \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= \frac {5}{2}+{\mathrm e}^{\frac {2 t}{5}} c_{1} \\ y \left (t \right ) &= -1-\frac {2 \,{\mathrm e}^{\frac {2 t}{5}} c_{1}}{5}-t \\ \end{align*}

Problem 2272

ODE	\begin {align*} x^{\prime }\left (t \right )&=3 x \left (t \right )\\ y^{\prime }\left (t \right )&=2 x \left (t \right )+3 y \left (t \right )\\ z^{\prime }\left (t \right )&=3 y \left (t \right )-2 z \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{3} {\mathrm e}^{3 t} \\ y \left (t \right ) &= {\mathrm e}^{3 t} \left (2 c_{3} t +c_{2} \right ) \\ z \left (t \right ) &= \left (\frac {3 \,{\mathrm e}^{5 t} \left (10 c_{3} t +5 c_{2} -2 c_{3} \right )}{25}+c_{1} \right ) {\mathrm e}^{-2 t} \\ \end{align*}

Problem 2273

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

Problem 2274

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

Problem 2275

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

Problem 2276

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

Problem 2277

ODE	\[ \boxed {x^{\prime \prime }-\frac {k^{2}}{x^{2}}=0} \]
program solution	\[ \frac {\sqrt {2}\, \left (\operatorname {arctanh}\left (\frac {\sqrt {-x \left (-c_{1} x+k^{2}\right )}}{x \sqrt {c_{1}}}\right ) k^{2}+\sqrt {-x \left (-c_{1} x+k^{2}\right )}\, \sqrt {c_{1}}\right )}{2 c_{1}^{\frac {3}{2}}} = t +c_{2} \] Verified OK. \[ -\frac {\sqrt {2}\, \left (\operatorname {arctanh}\left (\frac {\sqrt {-x \left (-c_{1} x+k^{2}\right )}}{x \sqrt {c_{1}}}\right ) k^{2}+\sqrt {-x \left (-c_{1} x+k^{2}\right )}\, \sqrt {c_{1}}\right )}{2 c_{1}^{\frac {3}{2}}} = c_{3} +t \] Verified OK.
Maple solution	\begin{align*} x \left (t \right ) &= \frac {c_{1} \left (c_{1}^{2} k^{4}+2 k^{2} c_{1} {\mathrm e}^{\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{4}-2 \textit {\_Z} \,c_{1}^{3} k^{2} {\mathrm e}^{\textit {\_Z}}-\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}-2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} -2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}+{\mathrm e}^{2 \operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{4}-2 \textit {\_Z} \,c_{1}^{3} k^{2} {\mathrm e}^{\textit {\_Z}}-\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}-2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} -2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}\right ) {\mathrm e}^{-\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{4}-2 \textit {\_Z} \,c_{1}^{3} k^{2} {\mathrm e}^{\textit {\_Z}}-\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}-2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} -2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}}{2} \\ x \left (t \right ) &= \frac {c_{1} \left (c_{1}^{2} k^{4}+2 k^{2} c_{1} {\mathrm e}^{\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{4}-2 \textit {\_Z} \,c_{1}^{3} k^{2} {\mathrm e}^{\textit {\_Z}}-\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}+2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} +2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}+{\mathrm e}^{2 \operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{4}-2 \textit {\_Z} \,c_{1}^{3} k^{2} {\mathrm e}^{\textit {\_Z}}-\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}+2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} +2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}\right ) {\mathrm e}^{-\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{4}-2 \textit {\_Z} \,c_{1}^{3} k^{2} {\mathrm e}^{\textit {\_Z}}-\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}+2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} +2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}}{2} \\ \end{align*}

Problem 2278

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

Problem 2279

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

Problem 2280

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

Problem 2281

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

Problem 2282

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

Problem 2283

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

Problem 2284

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

Problem 2285

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

Problem 2286

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

Problem 2287

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

Problem 2288

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

Problem 2289

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

Problem 2290

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

Problem 2291

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

Problem 2292

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

Problem 2293

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

Problem 2294

ODE	\[ \boxed {y y^{\prime \prime }-{y^{\prime }}^{2}=-1} \]
program solution	\[ y = \frac {\sinh \left (c_{2} \left (x +c_{3} \right ) {\mathrm e}^{c_{1}}\right ) {\mathrm e}^{-c_{1}}}{c_{2}} \] Verified OK. \[ y = -\frac {\sinh \left (c_{2} \left (x +c_{4} \right ) {\mathrm e}^{c_{1}}\right ) {\mathrm e}^{-c_{1}}}{c_{2}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {c_{1} \left ({\mathrm e}^{\frac {c_{2} +x}{c_{1}}}-{\mathrm e}^{\frac {-x -c_{2}}{c_{1}}}\right )}{2} \\ y \left (x \right ) &= \frac {c_{1} \left ({\mathrm e}^{\frac {c_{2} +x}{c_{1}}}-{\mathrm e}^{\frac {-x -c_{2}}{c_{1}}}\right )}{2} \\ \end{align*}

Problem 2295

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

Problem 2296

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

Problem 2297

ODE	\[ \boxed {2 y y^{\prime \prime }-{y^{\prime }}^{2}=0} \]
program solution	\[ y = \frac {1}{4} c_{1}^{2} c_{2}^{2}+\frac {1}{2} c_{1}^{2} c_{2} x +\frac {1}{4} c_{1}^{2} x^{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\left (c_{1} x +c_{2} \right )^{2}}{4} \\ \end{align*}

Problem 2298

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

Problem 2299

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }-{y^{\prime }}^{3}=0} \]
program solution	\[ y = x +c_{2} -\ln \left (-1-\sqrt {1+{\mathrm e}^{2 x +2 c_{2} +2 c_{1}}}\right ) \] Verified OK. \[ y = x +c_{2} -\ln \left (-1+\sqrt {1+{\mathrm e}^{2 x +2 c_{2} +2 c_{1}}}\right ) \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\operatorname {arctanh}\left (\sqrt {-c_{1} {\mathrm e}^{2 x}+1}\right )+c_{2} \\ y \left (x \right ) &= \operatorname {arctanh}\left (\sqrt {-c_{1} {\mathrm e}^{2 x}+1}\right )+c_{2} \\ \end{align*}

Problem 2300

ODE	\[ \boxed {\left (1+y\right ) y^{\prime \prime }-3 {y^{\prime }}^{2}=0} \]
program solution	\[ y = -\frac {\sqrt {-2 c_{1} c_{2} -2 c_{1} x}-1}{\sqrt {-2 c_{1} c_{2} -2 c_{1} x}} \] Verified OK. \[ y = -\frac {\sqrt {-2 c_{1} c_{2} -2 c_{1} x}+1}{\sqrt {-2 c_{1} c_{2} -2 c_{1} x}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -1 \\ y \left (x \right ) &= -\frac {\sqrt {-2 c_{1} x -2 c_{2}}-1}{\sqrt {-2 c_{1} x -2 c_{2}}} \\ y \left (x \right ) &= -\frac {\sqrt {-2 c_{1} x -2 c_{2}}+1}{\sqrt {-2 c_{1} x -2 c_{2}}} \\ \end{align*}