2.17.64 Problems 6301 to 6400

Problem 6301

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

Problem 6302

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }-10 y=6 \,{\mathrm e}^{4 x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-5 x}+\frac {c_{2} {\mathrm e}^{2 x}}{7}+\frac {{\mathrm e}^{4 x}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left ({\mathrm e}^{9 x}+3 \,{\mathrm e}^{7 x} c_{1} +3 c_{2} \right ) {\mathrm e}^{-5 x}}{3} \]

Problem 6303

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

Problem 6304

ODE	\[ \boxed {y^{\prime \prime }+10 y^{\prime }+25 y=14 \,{\mathrm e}^{-5 x}} \]
program solution	\[ y = {\mathrm e}^{-5 x} \left (c_{2} x +c_{1} \right )+7 x^{2} {\mathrm e}^{-5 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-5 x} \left (c_{1} x +7 x^{2}+c_{2} \right ) \]

Problem 6305

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+5 y=25 x^{2}+12} \]
program solution	\[ y = c_{1} {\mathrm e}^{x} \cos \left (2 x \right )+\frac {c_{2} {\mathrm e}^{x} \sin \left (2 x \right )}{2}+5 x^{2}+4 x +2 \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x} \sin \left (2 x \right ) c_{2} +{\mathrm e}^{x} \cos \left (2 x \right ) c_{1} +5 x^{2}+4 x +2 \]

Problem 6306

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }-6 y=20 \,{\mathrm e}^{-2 x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+\frac {c_{2} {\mathrm e}^{3 x}}{5}-4 \,{\mathrm e}^{-2 x} x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} {\mathrm e}^{5 x}+c_{1} -4 x \right ) {\mathrm e}^{-2 x} \]

Problem 6307

ODE	\[ \boxed {y^{\prime \prime }-3 y^{\prime }+2 y=14 \sin \left (2 x \right )-18 \cos \left (2 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{2 x}+3 \cos \left (2 x \right )+2 \sin \left (2 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{2 x} c_{1} +{\mathrm e}^{x} c_{2} +2 \sin \left (2 x \right )+3 \cos \left (2 x \right ) \]

Problem 6308

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

Problem 6309

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }=12 x -10} \]
program solution	\[ y = -3 x^{2}-\frac {c_{1}}{2}+2 x +1+c_{2} {\mathrm e}^{2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{2 x} c_{1}}{2}-3 x^{2}+2 x +c_{2} \]

Problem 6310

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

Problem 6311

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

Problem 6312

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }=10 x^{4}+2} \]
program solution	\[ y = 2 x^{5}-10 x^{4}+40 x^{3}-120 x^{2}+c_{1} +242 x -242+c_{2} {\mathrm e}^{-x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -c_{1} {\mathrm e}^{-x}-120 x^{2}+40 x^{3}-10 x^{4}+2 x^{5}+242 x +c_{2} \]

Problem 6313

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

Problem 6314

ODE	\[ \boxed {y^{\prime \prime }+9 y=2 \sin \left (3 x \right )+4 \sin \left (x \right )-26 \,{\mathrm e}^{-2 x}+27 x^{3}} \]
program solution	\[ y = c_{1} \cos \left (3 x \right )+\frac {c_{2} \sin \left (3 x \right )}{3}-2 \,{\mathrm e}^{-2 x}+\frac {\sin \left (x \right )}{2}-\frac {x \cos \left (3 x \right )}{3}-2 x +3 x^{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-2 x +6 c_{1} \right ) \cos \left (3 x \right )}{6}+\frac {\left (6 c_{2} +3\right ) \sin \left (3 x \right )}{6}+3 x^{3}-2 x +\frac {\sin \left (x \right )}{2}-2 \,{\mathrm e}^{-2 x} \]

Problem 6315

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

Problem 6316

ODE	\[ \boxed {y^{\prime \prime \prime }+y^{\prime }=\sin \left (x \right )} \]
program solution	\[ y = c_{1} +c_{2} {\mathrm e}^{i x}+{\mathrm e}^{-i x} c_{3} -\left (\int \sin \left (x \right )^{2}d x \right ) \sin \left (x \right )-\frac {\cos \left (x \right ) \left (\int \sin \left (2 x \right )d x +2\right )}{2} \] 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 6317

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

Problem 6318

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

Problem 6319

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

Problem 6320

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

Problem 6321

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

Problem 6322

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

Problem 6323

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

Problem 6324

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

Problem 6325

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

Problem 6326

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

Problem 6327

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

Problem 6328

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

Problem 6329

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

Problem 6330

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

Problem 6331

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

Problem 6332

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

Problem 6333

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

Problem 6334

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

Problem 6335

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

Problem 6336

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

Problem 6337

ODE	\[ \boxed {y^{\prime \prime }+y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \sin \left (x \right ) \end {align*}
program solution	\[ y = c_{1} \sin \left (x \right )-c_{2} \sin \left (x \right ) \cot \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sin \left (x \right )+\cos \left (x \right ) c_{2} \]

Problem 6338

ODE	\[ \boxed {y^{\prime \prime }-y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{x} \end {align*}
program solution	\[ y = c_{1} {\mathrm e}^{x}-\frac {c_{2} {\mathrm e}^{-2 x} {\mathrm e}^{x}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-x}+{\mathrm e}^{x} c_{2} \]

Problem 6339

ODE	\[ \boxed {x y^{\prime \prime }+3 y^{\prime }=0} \] Given that one solution of the ode is \begin {align*} y_1 &= 1 \end {align*}
program solution	\[ y = c_{1} -\frac {c_{2}}{2 x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} +\frac {c_{2}}{x^{2}} \]

Problem 6340

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x -4 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x^{2} \end {align*}
program solution	\[ y = c_{1} x^{2}-\frac {c_{2}}{4 x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{4}+c_{2}}{x^{2}} \]

Problem 6341

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +2 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x \end {align*}
program solution	\[ y = c_{1} x +c_{2} x \left (-\frac {\ln \left (1+x \right )}{2}+\frac {\ln \left (x -1\right )}{2}+\frac {1}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {c_{2} \ln \left (x +1\right ) x}{2}+\frac {c_{2} \ln \left (x -1\right ) x}{2}+c_{1} x +c_{2} \]

Problem 6342

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (-\frac {1}{4}+x^{2}\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \frac {\sin \left (x \right )}{\sqrt {x}} \end {align*}
program solution	\[ y = \frac {\sin \left (x \right ) c_{1}}{\sqrt {x}}-\frac {c_{2} \sin \left (x \right ) \cot \left (x \right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sin \left (x \right )+\cos \left (x \right ) c_{2}}{\sqrt {x}} \]

Problem 6343

ODE	\[ \boxed {y^{\prime \prime }-\frac {y^{\prime } x}{x -1}+\frac {y}{x -1}=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x \end {align*}
program solution	\[ y = c_{1} x +c_{2} {\mathrm e}^{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x +{\mathrm e}^{x} c_{2} \]

Problem 6344

ODE	\[ \boxed {x^{2} y^{\prime \prime }+2 y^{\prime } x -2 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x \end {align*}
program solution	\[ y = c_{1} x -\frac {c_{2}}{3 x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{3}+c_{1}}{x^{2}} \]

Problem 6345

ODE	\[ \boxed {x^{2} y^{\prime \prime }-x \left (x +2\right ) y^{\prime }+\left (x +2\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x \end {align*}
program solution	\[ y = c_{1} x +c_{2} x \,{\mathrm e}^{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x \left ({\mathrm e}^{x} c_{2} +c_{1} \right ) \]

Problem 6346

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

Problem 6347

ODE	\[ \boxed {x y^{\prime \prime }-\left (2 x +1\right ) y^{\prime }+y \left (1+x \right )=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{x} \end {align*}
program solution	\[ y = c_{1} {\mathrm e}^{x}+\frac {c_{2} x^{2} {\mathrm e}^{x}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x} \left (c_{2} x^{2}+c_{1} \right ) \]

Problem 6348

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

Problem 6349

ODE	\[ \boxed {y^{\prime \prime \prime }-3 y^{\prime \prime }+4 y^{\prime }-2 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{x}+{\mathrm e}^{\left (1+i\right ) x} c_{2} +{\mathrm e}^{\left (1-i\right ) x} c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x} \left (c_{1} +\sin \left (x \right ) c_{2} +c_{3} \cos \left (x \right )\right ) \]

Problem 6350

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

Problem 6351

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

Problem 6352

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

Problem 6353

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+6 y^{\prime \prime }+4 y^{\prime }+y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+c_{2} x \,{\mathrm e}^{-x}+c_{3} {\mathrm e}^{-x} x^{2}+x^{3} {\mathrm e}^{-x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-x} \left (c_{4} x^{3}+c_{3} x^{2}+c_{2} x +c_{1} \right ) \]

Problem 6354

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+c_{2} {\mathrm e}^{x}+{\mathrm e}^{i x} c_{3} +{\mathrm e}^{-i x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-x}+{\mathrm e}^{x} c_{2} +c_{3} \sin \left (x \right )+c_{4} \cos \left (x \right ) \]

Problem 6355

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+5 y^{\prime \prime }+4 y=0} \]
program solution	\[ y = {\mathrm e}^{i x} c_{1} +c_{2} {\mathrm e}^{-i x}+{\mathrm e}^{2 i x} c_{3} +{\mathrm e}^{-2 i x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2 c_{2} \cos \left (x \right )^{2}+\left (2 c_{1} \sin \left (x \right )+c_{4} \right ) \cos \left (x \right )+c_{3} \sin \left (x \right )-c_{2} \]

Problem 6356

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

Problem 6357

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

Problem 6358

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }+2 y^{\prime \prime }+2 y^{\prime }+y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+c_{2} x \,{\mathrm e}^{-x}+{\mathrm e}^{i x} c_{3} +{\mathrm e}^{-i x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-x} \left (c_{2} x +c_{1} \right )+c_{3} \sin \left (x \right )+c_{4} \cos \left (x \right ) \]

Problem 6359

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }-2 y^{\prime \prime }-6 y^{\prime }+5 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{x}+c_{2} x \,{\mathrm e}^{x}+{\mathrm e}^{\left (-2+i\right ) x} c_{3} +{\mathrm e}^{\left (-2-i\right ) x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} x +c_{1} \right ) {\mathrm e}^{3 x}+c_{3} \sin \left (x \right )+c_{4} \cos \left (x \right )\right ) {\mathrm e}^{-2 x} \]

Problem 6360

ODE	\[ \boxed {y^{\prime \prime \prime }-6 y^{\prime \prime }+11 y^{\prime }-6 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{2 x}+{\mathrm e}^{3 x} c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{2 x} c_{1} +c_{2} {\mathrm e}^{3 x}+c_{3} {\mathrm e}^{x} \]

Problem 6361

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

Problem 6362

ODE	\[ \boxed {y^{\left (5\right )}-6 y^{\prime \prime \prime \prime }-8 y^{\prime \prime \prime }+48 y^{\prime \prime }+16 y^{\prime }-96 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+c_{2} x \,{\mathrm e}^{-2 x}+{\mathrm e}^{2 x} c_{3} +x \,{\mathrm e}^{2 x} c_{4} +{\mathrm e}^{6 x} c_{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{5} x +c_{4} \right ) {\mathrm e}^{-2 x}+\left (c_{3} x +c_{2} \right ) {\mathrm e}^{2 x}+c_{1} {\mathrm e}^{6 x} \]

Problem 6363

ODE	\[ \boxed {y^{\prime \prime \prime \prime }=0} \]
program solution	\[ y = c_{4} x^{3}+c_{3} x^{2}+c_{2} x +c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {1}{6} c_{1} x^{3}+\frac {1}{2} c_{2} x^{2}+c_{3} x +c_{4} \]

Problem 6364

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

Problem 6365

ODE	\[ \boxed {y^{\prime \prime \prime }-3 y^{\prime \prime }+2 y^{\prime }=10+42 \,{\mathrm e}^{3 x}} \]
program solution	\[ y = c_{1} +c_{2} {\mathrm e}^{x}+{\mathrm e}^{2 x} c_{3} +5 x +7 \,{\mathrm e}^{3 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{2 x} c_{1}}{2}+{\mathrm e}^{x} c_{2} +7 \,{\mathrm e}^{3 x}+5 x +c_{3} \]

Problem 6366

ODE	\[ \boxed {y^{\prime \prime \prime }-y^{\prime }=1} \] With initial conditions \begin {align*} [y \left (0\right ) = 4, y^{\prime }\left (0\right ) = 4, y^{\prime \prime }\left (0\right ) = 4] \end {align*}
program solution	\[ y = -\frac {{\mathrm e}^{-x}}{2}+\frac {9 \,{\mathrm e}^{x}}{2}-x \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {{\mathrm e}^{-x}}{2}+\frac {9 \,{\mathrm e}^{x}}{2}-x \]

Problem 6367

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

Problem 6368

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

Problem 6369

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

Problem 6370

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

Problem 6371

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

Problem 6372

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

Problem 6373

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+9 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-3 x}+c_{2} x \,{\mathrm e}^{-3 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-3 x} \left (c_{2} x +c_{1} \right ) \]

Problem 6374

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }+6 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{\frac {x}{2}} \cos \left (\frac {x \sqrt {23}}{2}\right )+\frac {2 c_{2} \sin \left (\frac {x \sqrt {23}}{2}\right ) {\mathrm e}^{\frac {x}{2}} \sqrt {23}}{23} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{\frac {x}{2}} \left (c_{1} \sin \left (\frac {\sqrt {23}\, x}{2}\right )+c_{2} \cos \left (\frac {\sqrt {23}\, x}{2}\right )\right ) \]

Problem 6375

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }-5 y=x} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\left (-1+\sqrt {6}\right ) x}+\frac {c_{2} \sqrt {6}\, {\mathrm e}^{\left (1+\sqrt {6}\right ) x}}{12}-\frac {x}{5}+\frac {2}{25} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{\left (1+\sqrt {6}\right ) x} c_{2} +{\mathrm e}^{-\left (-1+\sqrt {6}\right ) x} c_{1} -\frac {x}{5}+\frac {2}{25} \]

Problem 6376

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

Problem 6377

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

Problem 6378

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

Problem 6379

ODE	\[ \boxed {y^{\prime \prime }+9 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 2] \end {align*}
program solution	\[ y = \frac {2 \sin \left (3 x \right )}{3}+\cos \left (3 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {2 \sin \left (3 x \right )}{3}+\cos \left (3 x \right ) \]

Problem 6380

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }+4 y=x} \] With initial conditions \begin {align*} [y \left (1\right ) = 2, y^{\prime }\left (1\right ) = 1] \end {align*}
program solution	\[ y = \frac {\left (\left (\sqrt {15}\, \sin \left (\frac {\sqrt {15}}{2}\right )+135 \cos \left (\frac {\sqrt {15}}{2}\right )\right ) \cos \left (\frac {\sqrt {15}\, x}{2}\right )-\sin \left (\frac {\sqrt {15}\, x}{2}\right ) \left (\sqrt {15}\, \cos \left (\frac {\sqrt {15}}{2}\right )-135 \sin \left (\frac {\sqrt {15}}{2}\right )\right )\right ) {\mathrm e}^{\frac {x}{2}-\frac {1}{2}}}{80}+\frac {x}{4}+\frac {1}{16} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\left (\sqrt {15}\, \sin \left (\frac {\sqrt {15}}{2}\right )+135 \cos \left (\frac {\sqrt {15}}{2}\right )\right ) \cos \left (\frac {\sqrt {15}\, x}{2}\right )-\sin \left (\frac {\sqrt {15}\, x}{2}\right ) \left (\sqrt {15}\, \cos \left (\frac {\sqrt {15}}{2}\right )-135 \sin \left (\frac {\sqrt {15}}{2}\right )\right )\right ) {\mathrm e}^{\frac {x}{2}-\frac {1}{2}}}{80}+\frac {x}{4}+\frac {1}{16} \]

Problem 6381

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+5 y={\mathrm e}^{x}} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {\left (-9 \cos \left (2 x \right )-\sin \left (2 x \right )\right ) {\mathrm e}^{-x}}{8}+\frac {{\mathrm e}^{x}}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-\sin \left (2 x \right )-9 \cos \left (2 x \right )\right ) {\mathrm e}^{-x}}{8}+\frac {{\mathrm e}^{x}}{8} \]

Problem 6382

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }+4 y=\sin \left (x \right )} \] With initial conditions \begin {align*} \left [y \left (\frac {\pi }{2}\right ) = 1, y^{\prime }\left (\frac {\pi }{2}\right ) = -1\right ] \end {align*}
program solution	\[ y = \frac {\left (\left (\sqrt {7}\, \sin \left (\frac {\sqrt {7}\, x}{2}\right )+35 \cos \left (\frac {\sqrt {7}\, x}{2}\right )\right ) \cos \left (\frac {\sqrt {7}\, \pi }{4}\right )-\sin \left (\frac {\sqrt {7}\, \pi }{4}\right ) \left (\sqrt {7}\, \cos \left (\frac {\sqrt {7}\, x}{2}\right )-35 \sin \left (\frac {\sqrt {7}\, x}{2}\right )\right )\right ) {\mathrm e}^{-\frac {3 x}{2}+\frac {3 \pi }{4}}}{42}-\frac {\cos \left (x \right )}{6}+\frac {\sin \left (x \right )}{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\left (\sqrt {7}\, \sin \left (\frac {\sqrt {7}\, x}{2}\right )+35 \cos \left (\frac {\sqrt {7}\, x}{2}\right )\right ) \cos \left (\frac {\sqrt {7}\, \pi }{4}\right )-\sin \left (\frac {\sqrt {7}\, \pi }{4}\right ) \left (\sqrt {7}\, \cos \left (\frac {\sqrt {7}\, x}{2}\right )-35 \sin \left (\frac {\sqrt {7}\, x}{2}\right )\right )\right ) {\mathrm e}^{-\frac {3 x}{2}+\frac {3 \pi }{4}}}{42}-\frac {\cos \left (x \right )}{6}+\frac {\sin \left (x \right )}{6} \]

Problem 6383

ODE	\[ \boxed {y^{\prime \prime }+y={\mathrm e}^{-x}} \] With initial conditions \begin {align*} [y \left (2\right ) = 0, y^{\prime }\left (2\right ) = -2] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{-x}}{2}+\frac {\left (\left (-\cos \left (x \right )+\sin \left (x \right )\right ) \cos \left (2\right )-\cos \left (x \right ) \sin \left (2\right )-\sin \left (x \right ) \sin \left (2\right )\right ) {\mathrm e}^{-2}}{2}-2 \sin \left (x \right ) \cos \left (2\right )+2 \cos \left (x \right ) \sin \left (2\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{-x}}{2}+\frac {\left (\left (-\cos \left (x \right )+\sin \left (x \right )\right ) \cos \left (2\right )-\sin \left (2\right ) \cos \left (x \right )-\sin \left (2\right ) \sin \left (x \right )\right ) {\mathrm e}^{-2}}{2}-2 \cos \left (2\right ) \sin \left (x \right )+2 \sin \left (2\right ) \cos \left (x \right ) \]

Problem 6384

ODE	\[ \boxed {y^{\prime \prime }-y=\cos \left (x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (2\right ) = 2] \end {align*}
program solution	\[ y = \frac {\left (\sin \left (2\right )-4\right ) {\mathrm e}^{-x +2}+\left (\sin \left (2\right )-4\right ) {\mathrm e}^{-x +6}+7 \,{\mathrm e}^{4-x}+7 \,{\mathrm e}^{8-x}+\left (4-\sin \left (2\right )\right ) {\mathrm e}^{x +2}+\left (4-\sin \left (2\right )\right ) {\mathrm e}^{x +6}+7 \,{\mathrm e}^{x +4}+\left (-2 \,{\mathrm e}^{4}-{\mathrm e}^{8}-1\right ) \cos \left (x \right )+7 \,{\mathrm e}^{x}}{2 \,{\mathrm e}^{8}+4 \,{\mathrm e}^{4}+2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\sin \left (2\right )-4\right ) {\mathrm e}^{2-x}+7 \,{\mathrm e}^{4-x}+\left (-\sin \left (2\right )+4\right ) {\mathrm e}^{x +2}+\left (-{\mathrm e}^{4}-1\right ) \cos \left (x \right )+7 \,{\mathrm e}^{x}}{2 \,{\mathrm e}^{4}+2} \]

Problem 6385

ODE	\[ \boxed {y^{\prime \prime }=\tan \left (x \right )} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = -1] \end {align*}
program solution	\[ y = \frac {\left (-i-i {\mathrm e}^{2 i}\right ) \operatorname {polylog}\left (2, -{\mathrm e}^{2 i x}\right )+2 x \left ({\mathrm e}^{2 i}+1\right ) \ln \left ({\mathrm e}^{2 i x}+1\right )+\left (i {\mathrm e}^{2 i}+i\right ) \operatorname {dilog}\left ({\mathrm e}^{2 i}+1\right )+\left (-2 \,{\mathrm e}^{2 i}-2\right ) \ln \left ({\mathrm e}^{2 i}+1\right )+\left (2 \ln \left (\cos \left (1\right )\right ) x -2 x \ln \left (\cos \left (x \right )\right )+\left (2-2 x \right ) \tan \left (1\right )-i x^{2}+\left (-2-2 i\right ) x +4+3 i\right ) {\mathrm e}^{2 i}+2 \ln \left (\cos \left (1\right )\right ) x -2 x \ln \left (\cos \left (x \right )\right )+\left (2-2 x \right ) \tan \left (1\right )-i x^{2}+\left (-2+2 i\right ) x +4-i}{2 \,{\mathrm e}^{2 i}+2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-i {\mathrm e}^{2 i}-i\right ) \operatorname {polylog}\left (2, -{\mathrm e}^{2 i x}\right )+2 x \left ({\mathrm e}^{2 i}+1\right ) \ln \left ({\mathrm e}^{2 i x}+1\right )+\left (i {\mathrm e}^{2 i}+i\right ) \operatorname {polylog}\left (2, -{\mathrm e}^{2 i}\right )+\left (-2 \,{\mathrm e}^{2 i}-2\right ) \ln \left ({\mathrm e}^{2 i}+1\right )+\left (2 \ln \left (\cos \left (1\right )\right ) x -2 x \ln \left (\cos \left (x \right )\right )+\left (-2 x +2\right ) \tan \left (1\right )-i x^{2}+\left (-2-2 i\right ) x +4+3 i\right ) {\mathrm e}^{2 i}+2 \ln \left (\cos \left (1\right )\right ) x -2 x \ln \left (\cos \left (x \right )\right )+\left (-2 x +2\right ) \tan \left (1\right )-i x^{2}+\left (-2+2 i\right ) x +4-i}{2 \,{\mathrm e}^{2 i}+2} \]

Problem 6386

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }=\ln \left (x \right )} \] With initial conditions \begin {align*} [y \left (1\right ) = {\mathrm e}, y^{\prime }\left (1\right ) = {\mathrm e}^{-1}] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{2 x} \operatorname {expIntegral}_{1}\left (2\right )}{4}+\frac {{\mathrm e}^{2 x -3}}{2}+{\mathrm e}-\frac {1}{2}-\frac {{\mathrm e}^{-1}}{2}-\frac {{\mathrm e}^{2 x} \operatorname {expIntegral}_{1}\left (2 x \right )}{4}-\frac {\ln \left (x \right ) x}{2}-\frac {\ln \left (x \right )}{4}+\frac {x}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\int _{1}^{x}\left ({\mathrm e}^{2 \textit {\_z1}} \operatorname {expIntegral}_{1}\left (2\right )-{\mathrm e}^{2 \textit {\_z1}} \operatorname {expIntegral}_{1}\left (2 \textit {\_z1} \right )+2 \,{\mathrm e}^{2 \textit {\_z1} -3}-\ln \left (\textit {\_z1} \right )\right )d \textit {\_z1} \right )}{2}+{\mathrm e} \]

Problem 6387

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

Problem 6388

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

Problem 6389

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

Problem 6390

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }-y=\sin \left (x \right ) {\mathrm e}^{x} x} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\left (1+\sqrt {2}\right ) x}+\frac {c_{2} \sqrt {2}\, {\mathrm e}^{\left (\sqrt {2}-1\right ) x}}{4}+\frac {62 \,{\mathrm e}^{x} \cos \left (x \right )}{289}+\frac {44 \sin \left (x \right ) {\mathrm e}^{x}}{289}-\frac {4 \,{\mathrm e}^{x} \cos \left (x \right ) x}{17}+\frac {\sin \left (x \right ) {\mathrm e}^{x} x}{17} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\left (1+\sqrt {2}\right ) x} c_{1} +{\mathrm e}^{\left (\sqrt {2}-1\right ) x} c_{2} -\frac {4 \,{\mathrm e}^{x} \left (\left (x -\frac {31}{34}\right ) \cos \left (x \right )-\frac {\sin \left (x \right ) \left (x +\frac {44}{17}\right )}{4}\right )}{17} \]

Problem 6391

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

Problem 6392

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

Problem 6393

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

Problem 6394

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

Problem 6395

ODE	\[ \boxed {y^{\prime \prime }-y=3 \,{\mathrm e}^{2 x}} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{2 x} \end {align*}
program solution	\[ y = c_{1} {\mathrm e}^{2 x}-\frac {c_{2} {\mathrm e}^{-2 x}}{4}+\frac {3 \,{\mathrm e}^{2 x} \left (-1+4 x \right )}{16} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} {\mathrm e}^{-x}+{\mathrm e}^{x} c_{1} +{\mathrm e}^{2 x} \]

Problem 6396

ODE	\[ \boxed {y^{\prime \prime }+y=-8 \sin \left (3 x \right )} \] Given that one solution of the ode is \begin {align*} y_1 &= \sin \left (3 x \right ) \end {align*}
program solution	\[ y = c_{1} \sin \left (3 x \right )-\frac {c_{2} \cos \left (3 x \right )}{3}-\frac {2 \sin \left (3 x \right )}{9}+\frac {4 x \cos \left (3 x \right )}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sin \left (x \right ) c_{2} +\cos \left (x \right ) c_{1} +\sin \left (3 x \right ) \]

Problem 6397

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }+y=x^{2}+2 x +2} \] Given that one solution of the ode is \begin {align*} y_1 &= x^{2} \end {align*}
program solution	\[ y = \frac {-c_{2} \left (x^{2}-x +2\right ) {\mathrm e}^{-x}+6 \left (\frac {\operatorname {expIntegral}_{1}\left (x \right ) c_{2}}{6}+c_{1} \right ) x^{3}}{6 x}+\frac {-\left (\int _{0}^{x}\frac {\left (\alpha ^{2}+2 \alpha +2\right ) \left (\operatorname {expIntegral}_{1}\left (\alpha \right ) {\mathrm e}^{\alpha } \alpha ^{3}-\alpha ^{2}+\alpha -2\right )}{\alpha }d \alpha \right ) x^{3}+\left (x^{4}-2 x^{3}+8 x^{2}-16 x +16\right ) \left (\operatorname {expIntegral}_{1}\left (x \right ) {\mathrm e}^{x} x^{3}-x^{2}+x -2\right )}{6 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )+{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) c_{1} +x^{2} \]

Problem 6398

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }=\frac {x -1}{x}} \] Given that one solution of the ode is \begin {align*} y_1 &= \ln \left (x \right ) \end {align*}
program solution	\[ y = -\ln \left (x \right ) \left (\int \frac {{\mathrm e}^{-x} \left (x -1\right )}{\ln \left (x \right )}d x \right ) c_{2} -c_{2} x \,{\mathrm e}^{-x}+c_{1} \ln \left (x \right )-\ln \left (x \right ) \left (\int \frac {{\mathrm e}^{-x} \left (x -1\right )}{\ln \left (x \right )}d x \right ) \left (\int _{0}^{x}\frac {\ln \left (\alpha \right ) \left (\alpha -1\right ) {\mathrm e}^{\alpha }}{\alpha }d \alpha \right )-{\mathrm e}^{-x} \left (\int _{0}^{x}\frac {\ln \left (\alpha \right ) \left (\alpha -1\right ) {\mathrm e}^{\alpha }}{\alpha }d \alpha \right ) x +\left (\int _{0}^{x}\frac {\left (\alpha -1\right ) \left (\left (\int \frac {{\mathrm e}^{-\alpha } \left (\alpha -1\right )}{\ln \left (\alpha \right )}d \alpha \right ) {\mathrm e}^{\alpha } \ln \left (\alpha \right )+\alpha \right )}{\alpha }d \alpha \right ) \ln \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \int \left (1+{\mathrm e}^{-x} \operatorname {expIntegral}_{1}\left (-x \right )+c_{1} {\mathrm e}^{-x}\right )d x +c_{2} \]

Problem 6399

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

Problem 6400

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