2.17.114 Problems 11301 to 11400

Problem 11301

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

Problem 11302

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

Problem 11303

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

Problem 11304

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

Problem 11305

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

Problem 11306

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

Problem 11307

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

Problem 11308

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

Problem 11309

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

Problem 11310

ODE	\[ \boxed {x^{4} y^{\prime \prime }+2 x^{3} \left (x +1\right ) y^{\prime }+y n^{2}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \operatorname {HeunD}\left (8 \left (-n^{2}\right )^{\frac {1}{4}}, \frac {-8 i \left (-n^{2}\right )^{\frac {3}{4}}-n +8 \sqrt {-n^{2}}\, n}{n}, -\frac {16 i \left (-n^{2}\right )^{\frac {3}{4}}}{n}, \frac {n -8 i \left (-n^{2}\right )^{\frac {3}{4}}-8 \sqrt {-n^{2}}\, n}{n}, \frac {\left (-n^{2}\right )^{\frac {1}{4}} x -i n}{\left (-n^{2}\right )^{\frac {1}{4}} x +i n}\right ) {\mathrm e}^{\frac {i \sqrt {-n^{2}}\, x^{2}+i n^{2}-n \,x^{2}}{x n}}+c_{2} \operatorname {HeunD}\left (-8 \left (-n^{2}\right )^{\frac {1}{4}}, \frac {-8 i \left (-n^{2}\right )^{\frac {3}{4}}-n +8 \sqrt {-n^{2}}\, n}{n}, -\frac {16 i \left (-n^{2}\right )^{\frac {3}{4}}}{n}, \frac {n -8 i \left (-n^{2}\right )^{\frac {3}{4}}-8 \sqrt {-n^{2}}\, n}{n}, \frac {\left (-n^{2}\right )^{\frac {1}{4}} x -i n}{\left (-n^{2}\right )^{\frac {1}{4}} x +i n}\right ) {\mathrm e}^{\frac {-i \sqrt {-n^{2}}\, x^{2}-i n^{2}-n \,x^{2}}{x n}}}{\sqrt {x}} \]

Problem 11311

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	\[ y \left (x \right ) = \frac {\ln \left (c_{1} x -1\right ) c_{1}^{2}+c_{2} c_{1}^{2}+c_{1} x +\ln \left (c_{1} x -1\right )}{c_{1}^{2}} \]

Problem 11312

ODE
program solution	N/A
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (x^{2}+2\right ) \sqrt {-x^{2}+1}}{6}+c_{1} x +\frac {x \arcsin \left (x \right )}{2}+c_{2} \\ y \left (x \right ) &= -\frac {x^{2} \sqrt {-x^{2}+1}}{6}-\frac {\sqrt {-x^{2}+1}}{3}-\frac {x \arcsin \left (x \right )}{2}+c_{1} x +c_{2} \\ y \left (x \right ) &= \frac {\sqrt {c_{1}^{2}-1}\, x^{3}}{6}+\frac {c_{1} x^{2}}{2}+c_{2} x +c_{3} \\ \end{align*}

Problem 11313

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

Problem 11314

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

Problem 11315

ODE	\[ \boxed {\left (y^{\prime }-x y^{\prime \prime }\right )^{2}-{y^{\prime \prime }}^{2}=1} \]
program solution	\[ y = \frac {x \left (c_{1} x +2 \sqrt {c_{1}^{2}+1}\right )}{2}+c_{3} \] Verified OK. \[ y = \frac {\sqrt {-\frac {1}{x^{2}-1}}\, \sqrt {x^{2}-1}\, \left (-x \sqrt {x^{2}-1}+\ln \left (x +\sqrt {x^{2}-1}\right )\right )}{2}+c_{4} \] Verified OK. \[ y = \frac {c_{2} x^{2}}{2}-\sqrt {c_{2}^{2}+1}\, x +c_{5} \] Verified OK. \[ y = -\frac {\sqrt {-\frac {1}{x^{2}-1}}\, \sqrt {x^{2}-1}\, \left (-x \sqrt {x^{2}-1}+\ln \left (x +\sqrt {x^{2}-1}\right )\right )}{2}+c_{6} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {x \sqrt {-x^{2}+1}}{2}+\frac {\arcsin \left (x \right )}{2}+c_{1} \\ y \left (x \right ) &= -\frac {x \sqrt {-x^{2}+1}}{2}-\frac {\arcsin \left (x \right )}{2}+c_{1} \\ y \left (x \right ) &= \frac {\sqrt {c_{1}^{2}-1}\, x^{2}}{2}+c_{1} x +c_{2} \\ \end{align*}

Problem 11316

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

Problem 11317

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 {-c_{2} -x}{c_{1}}}\right )}{2} \\ y \left (x \right ) &= -\frac {c_{1} \left (-{\mathrm e}^{\frac {c_{2} +x}{c_{1}}}+{\mathrm e}^{\frac {-c_{2} -x}{c_{1}}}\right )}{2} \\ \end{align*}

Problem 11318

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

Problem 11319

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

Problem 11320

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

Problem 11321

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

Problem 11322

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

Problem 11323

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

Problem 11324

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

Problem 11325

ODE	\[ \boxed {\left (x^{3}-x \right ) y^{\prime \prime \prime }+\left (8 x^{2}-3\right ) y^{\prime \prime }+14 y^{\prime } x +4 y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \frac {\frac {c_{3}}{\sqrt {1+x}\, \sqrt {-1+x}}+c_{1} +\frac {c_{2} \ln \left (x +\sqrt {x^{2}-1}\right )}{\sqrt {x^{2}-1}}}{x} \]

Problem 11326

ODE
program solution	N/A
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\sqrt {-x \left (c_{1} x^{2}+2 c_{2} x -2 c_{3} \right )}}{x^{2}} \\ y \left (x \right ) &= -\frac {\sqrt {-x \left (c_{1} x^{2}+2 c_{2} x -2 c_{3} \right )}}{x^{2}} \\ \end{align*}

Problem 11327

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

Problem 11328

ODE	\[ \boxed {x^{2} \left (-x^{3}+1\right ) y^{\prime \prime }-x^{3} y^{\prime }-2 y=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 11329

ODE	\[ \boxed {x^{2} y^{\prime \prime \prime }-5 x y^{\prime \prime }+\left (4 x^{4}+5\right ) y^{\prime }-8 y x^{3}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = c_{1} x^{2}+c_{2} \cos \left (x^{2}\right )+c_{3} \sin \left (x^{2}\right ) \]

Problem 11330

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

Problem 11331

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

Problem 11332

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

Problem 11333

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

Problem 11334

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

Problem 11335

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	\[ y \left (x \right ) = -\ln \left (c_{1} \sin \left (x \right )-c_{2} \cos \left (x \right )\right ) \]

Problem 11336

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

Problem 11337

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 11338

ODE	\[ \boxed {\left (x^{3}+1\right ) y^{\prime \prime \prime }+9 x^{2} y^{\prime \prime }+18 y^{\prime } x +6 y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{2}+c_{2} x +c_{3}}{\left (1+x \right ) \left (x^{2}-x +1\right )} \]

Problem 11339

ODE	\[ \boxed {\left (x^{2}-x \right ) y^{\prime \prime }+\left (4 x +2\right ) y^{\prime }+2 y=0} \]
program solution	\[ y = \frac {-12 \ln \left (x \right ) c_{1} x^{3}+\left (3 x^{4}-18 x^{2}+6 x -1\right ) c_{1} +3 c_{2} x^{3}}{3 \left (x -1\right )^{5}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {12 \ln \left (x \right ) c_{1} x^{3}+\left (-3 x^{4}+18 x^{2}-6 x +1\right ) c_{1} +c_{2} x^{3}}{\left (-1+x \right )^{5}} \]

Problem 11340

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

Problem 11341

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

Problem 11342

ODE	\[ \boxed {x \left (x +2 y\right ) y^{\prime \prime }+2 x {y^{\prime }}^{2}+4 \left (y+x \right ) y^{\prime }+2 y=-x^{2}} \]
program solution	\[ \frac {x \left (x^{3}+12 x y+12 y^{2}-12 c_{1} \right )}{12} = c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {-3 x^{2}+\sqrt {3}\, \sqrt {-x \left (x^{4}-3 x^{3}+12 c_{2} x -12 c_{1} \right )}}{6 x} \\ y \left (x \right ) &= \frac {-3 x^{2}-\sqrt {3}\, \sqrt {-x \left (x^{4}-3 x^{3}+12 c_{2} x -12 c_{1} \right )}}{6 x} \\ \end{align*}

Problem 11343

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

Problem 11344

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

Problem 11345

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

Problem 11346

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

Problem 11347

ODE	\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )-\frac {2 y \left (t \right )}{3}+\frac {{\mathrm e}^{t}}{3}\\ y^{\prime }\left (t \right )&=\frac {4 x \left (t \right )}{3}+y \left (t \right )-t \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= -\frac {{\mathrm e}^{\frac {t}{3}} c_{2}}{2}-{\mathrm e}^{-\frac {t}{3}} c_{1} -6 t \\ y \left (t \right ) &= {\mathrm e}^{\frac {t}{3}} c_{2} +{\mathrm e}^{-\frac {t}{3}} c_{1} +9 t +9+\frac {{\mathrm e}^{t}}{2} \\ \end{align*}

Problem 11348

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

Problem 11349

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

Problem 11350

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

Problem 11351

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

Problem 11352

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

Problem 11353

ODE	\[ \boxed {x^{\prime }+2 x=t^{2}+4 t +7} \]
program solution	\[ x = \frac {\left (2 \,{\mathrm e}^{2 t} t^{2}+6 \,{\mathrm e}^{2 t} t +11 \,{\mathrm e}^{2 t}+4 c_{1} \right ) {\mathrm e}^{-2 t}}{4} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {t^{2}}{2}+\frac {3 t}{2}+\frac {11}{4}+{\mathrm e}^{-2 t} c_{1} \]

Problem 11354

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

Problem 11355

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

Problem 11356

ODE	\[ \boxed {2 x^{\prime \prime }-5 x^{\prime }-3 x=0} \]
program solution	\[ x = c_{1} {\mathrm e}^{-\frac {t}{2}}+\frac {2 c_{2} {\mathrm e}^{3 t}}{7} \] Verified OK.
Maple solution	\[ x \left (t \right ) = c_{1} {\mathrm e}^{3 t}+c_{2} {\mathrm e}^{-\frac {t}{2}} \]

Problem 11357

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

Problem 11358

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

Problem 11359

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

Problem 11360

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

Problem 11361

ODE	\[ \boxed {x^{\prime \prime }=-3 \sqrt {t}} \] With initial conditions \begin {align*} [x \left (1\right ) = 4, x^{\prime }\left (1\right ) = 2] \end {align*}
program solution	\[ x = -\frac {4 t^{\frac {5}{2}}}{5}+4 t +\frac {4}{5} \] Verified OK.
Maple solution	\[ x \left (t \right ) = -\frac {4 t^{\frac {5}{2}}}{5}+4 t +\frac {4}{5} \]

Problem 11362

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

Problem 11363

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

Problem 11364

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

Problem 11365

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

Problem 11366

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

Problem 11367

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

Problem 11368

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

Problem 11369

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

Problem 11370

ODE	\[ \boxed {u^{\prime }-\frac {1}{5-2 u}=0} \]
program solution	\[ u = \frac {5}{2}-\frac {\sqrt {25-4 t -4 c_{1}}}{2} \] Verified OK. \[ u = \frac {5}{2}+\frac {\sqrt {25-4 t -4 c_{1}}}{2} \] Verified OK.
Maple solution	\begin{align*} u \left (t \right ) &= \frac {5}{2}-\frac {\sqrt {25-4 t -4 c_{1}}}{2} \\ u \left (t \right ) &= \frac {5}{2}+\frac {\sqrt {25-4 t -4 c_{1}}}{2} \\ \end{align*}

Problem 11371

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

Problem 11372

ODE	\[ \boxed {Q^{\prime }-\frac {Q}{4+Q^{2}}=0} \]
program solution	\[ Q = {\mathrm e}^{-\frac {\operatorname {LambertW}\left (\frac {{\mathrm e}^{\frac {c_{1}}{2}+\frac {t}{2}}}{4}\right )}{2}+\frac {t}{4}+\frac {c_{1}}{4}} \] Verified OK.
Maple solution	\[ Q \left (t \right ) = \frac {2 \,{\mathrm e}^{\frac {t}{4}+\frac {c_{1}}{4}}}{\sqrt {\frac {{\mathrm e}^{\frac {t}{2}+\frac {c_{1}}{2}}}{\operatorname {LambertW}\left (\frac {{\mathrm e}^{\frac {t}{2}+\frac {c_{1}}{2}}}{4}\right )}}} \]

Problem 11373

ODE	\[ \boxed {x^{\prime }-{\mathrm e}^{x^{2}}=0} \]
program solution	\[ \int _{}^{x}{\mathrm e}^{-\textit {\_a}^{2}}d \textit {\_a} = t +c_{1} \] Verified OK.
Maple solution	\[ t -\frac {\sqrt {\pi }\, \operatorname {erf}\left (x \left (t \right )\right )}{2}+c_{1} = 0 \]

Problem 11374

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

Problem 11375

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

Problem 11376

ODE	\[ \boxed {\theta ^{\prime }-t \sqrt {t^{2}+1}\, \sec \left (\theta \right )=0} \]
program solution	\[ -\frac {\left (t^{2}+1\right )^{\frac {3}{2}}}{3}+\sin \left (\theta \right ) = c_{1} \] Verified OK.
Maple solution	\[ \theta \left (t \right ) = \arcsin \left (\frac {t^{2} \sqrt {t^{2}+1}}{3}+\frac {\sqrt {t^{2}+1}}{3}+c_{1} \right ) \]

Problem 11377

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

Problem 11378

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

Problem 11379

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

Problem 11380

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

Problem 11381

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

Problem 11382

ODE	\[ \boxed {x^{\prime }-\left (4 t -x\right )^{2}=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 1] \end {align*}
program solution	\[ x = \frac {4 \,{\mathrm e}^{-2 t \left (t -1\right )} t +12 \,{\mathrm e}^{-2 t \left (1+t \right )} t -2 \,{\mathrm e}^{-2 t \left (t -1\right )}+6 \,{\mathrm e}^{-2 t \left (1+t \right )}}{{\mathrm e}^{-2 t \left (t -1\right )}+3 \,{\mathrm e}^{-2 t \left (1+t \right )}} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {\left (4 t -2\right ) {\mathrm e}^{4 t}+12 t +6}{3+{\mathrm e}^{4 t}} \]

Problem 11383

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

Problem 11384

ODE	\[ \boxed {x^{\prime }-t^{2} {\mathrm e}^{-x}=0} \] With initial conditions \begin {align*} [x \left (0\right ) = \ln \left (2\right )] \end {align*}
program solution	\[ x = -\ln \left (3\right )+\ln \left (t^{3}+6\right ) \] Verified OK.
Maple solution	\[ x \left (t \right ) = -\ln \left (3\right )+\ln \left (t^{3}+6\right ) \]

Problem 11385

ODE	\[ \boxed {x^{\prime }-x \left (4+x\right )=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 1] \end {align*}
program solution	\[ \frac {\ln \left (x\right )}{4}-\frac {\ln \left (4+x\right )}{4} = t -\frac {\ln \left (5\right )}{4} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {4}{-1+5 \,{\mathrm e}^{-4 t}} \]

Problem 11386

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

Problem 11387

ODE	\[ \boxed {T^{\prime }-2 a t \left (T^{2}-a^{2}\right )=0} \] With initial conditions \begin {align*} [T \left (0\right ) = 0] \end {align*}
program solution	\[ T = -\frac {\sinh \left (t^{2} a^{2}\right ) a}{\cosh \left (t^{2} a^{2}\right )} \] Verified OK.
Maple solution	\[ T \left (t \right ) = -\frac {a \left ({\mathrm e}^{2 t^{2} a^{2}}-1\right )}{{\mathrm e}^{2 t^{2} a^{2}}+1} \]

Problem 11388

ODE	\[ \boxed {y^{\prime }-t^{2} \tan \left (y\right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	N/A
Maple solution	\[ y \left (t \right ) = 0 \]

Problem 11389

ODE	\[ \boxed {x^{\prime }-\frac {\left (4+2 t \right ) x}{\ln \left (x\right )}=0} \] With initial conditions \begin {align*} [x \left (0\right ) = {\mathrm e}] \end {align*}
program solution	\[ -\frac {t^{2}}{2}-2 t +\frac {\ln \left (x\right )^{2}}{4} = {\frac {1}{4}} \] Verified OK.
Maple solution	\[ x \left (t \right ) = {\mathrm e}^{\sqrt {2 t^{2}+8 t +1}} \]

Problem 11390

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

Problem 11391

ODE	\[ \boxed {x^{\prime }-\frac {t^{2}}{1-x^{2}}=0} \] With initial conditions \begin {align*} [x \left (1\right ) = 1] \end {align*}
program solution	\[ -\frac {t^{3}}{3}-\frac {x^{3}}{3}+x = {\frac {1}{3}} \] Verified OK.
Maple solution	\begin{align*} x \left (t \right ) &= \frac {\left (-4-4 t^{3}+4 \sqrt {t^{6}+2 t^{3}-3}\right )^{\frac {2}{3}}+4}{2 \left (-4-4 t^{3}+4 \sqrt {t^{6}+2 t^{3}-3}\right )^{\frac {1}{3}}} \\ x \left (t \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (-4-4 t^{3}+4 \sqrt {t^{6}+2 t^{3}-3}\right )^{\frac {2}{3}}-4 i \sqrt {3}+4}{4 \left (-4-4 t^{3}+4 \sqrt {t^{6}+2 t^{3}-3}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 11392

ODE	\[ \boxed {x^{\prime }-6 t \left (x-1\right )^{\frac {2}{3}}=0} \]
program solution	\[ x = t^{6}+6 t^{4} c_{1} +12 t^{2} c_{1}^{2}+8 c_{1}^{3}+1 \] Verified OK.
Maple solution	\[ c_{1} +t^{2}-\left (x \left (t \right )-1\right )^{\frac {1}{3}} = 0 \]

Problem 11393

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

Problem 11394

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

Problem 11395

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

Problem 11396

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

Problem 11397

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

Problem 11398

ODE	\[ \boxed {x^{\prime }-2 t^{3} x=-6} \]
program solution	\[ x = -\frac {{\mathrm e}^{\frac {t^{4}}{2}} \left (24 \,2^{\frac {1}{8}} {\mathrm e}^{-\frac {t^{4}}{4}} \operatorname {WhittakerM}\left (\frac {1}{8}, \frac {5}{8}, \frac {t^{4}}{2}\right ) t +30 \,{\mathrm e}^{-\frac {t^{4}}{2}} \left (t^{4}\right )^{\frac {1}{8}} t -5 c_{1} \left (t^{4}\right )^{\frac {1}{8}}\right )}{5 \left (t^{4}\right )^{\frac {1}{8}}} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {-\frac {24 \operatorname {WhittakerM}\left (\frac {1}{8}, \frac {5}{8}, \frac {t^{4}}{2}\right ) {\mathrm e}^{\frac {t^{4}}{4}} 2^{\frac {1}{8}} t}{5}+\left (t^{4}\right )^{\frac {1}{8}} \left ({\mathrm e}^{\frac {t^{4}}{2}} c_{1} -6 t \right )}{\left (t^{4}\right )^{\frac {1}{8}}} \]

Problem 11399

ODE	\[ \boxed {\cos \left (t \right ) x^{\prime }-2 x \sin \left (x\right )=0} \]
program solution	\[ -\ln \left (\sec \left (t \right )+\tan \left (t \right )\right )+\int _{0}^{x}\frac {\csc \left (\textit {\_a} \right )}{2 \textit {\_a}}d \textit {\_a} = c_{1} \] Verified OK.
Maple solution	\[ \ln \left (\sec \left (t \right )+\tan \left (t \right )\right )-\frac {\left (\int _{}^{x \left (t \right )}\frac {\csc \left (\textit {\_a} \right )}{\textit {\_a}}d \textit {\_a} \right )}{2}+c_{1} = 0 \]

Problem 11400

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