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# |
ODE |
solution |
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\[ {}f \left (x \right ) \sin \left (x \right ) y x y^{\prime } \pi = 0 \] |
\[ y = c_{1} \] Verified OK. |
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\[ {}y y^{\prime } = 0 \] |
\[ y = c_{1} \] Verified OK. |
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\[ {}x y y^{\prime } = 0 \] |
\[ y = c_{1} \] Verified OK. |
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\[ {}x y \sin \left (x \right ) y^{\prime } = 0 \] |
\[ y = c_{1} \] Verified OK. |
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\[ {}\pi y \sin \left (x \right ) y^{\prime } = 0 \] |
\[ y = c_{1} \] Verified OK. |
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\[ {}y {y^{\prime }}^{2} = 0 \] |
\[ y = c_{1} \] Verified OK. \[ y = c_{2} \] Verified OK. |
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\[ {}y^{3} {y^{\prime \prime }}^{2}+y y^{\prime } = 0 \] |
\[ \int _{}^{y}-\frac {4}{\left (12 \ln \left (\textit {\_a} \right )-12 c_{1} \right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{3} \] Verified OK. \[ \int _{}^{y}-\frac {16}{\left (12 \ln \left (\textit {\_a} \right )-12 c_{1} \right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}d \textit {\_a} = x +c_{4} \] Verified OK. \[ \int _{}^{y}-\frac {16}{\left (12 \ln \left (\textit {\_a} \right )-12 c_{1} \right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}d \textit {\_a} = x +c_{5} \] Verified OK. \[ \int _{}^{y}-\frac {4}{\left (-12 \ln \left (\textit {\_a} \right )-12 c_{2} \right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{6} \] Verified OK. \[ \int _{}^{y}-\frac {16}{\left (-12 \ln \left (\textit {\_a} \right )-12 c_{2} \right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )^{2}}d \textit {\_a} = x +c_{7} \] Verified OK. \[ \int _{}^{y}-\frac {16}{\left (-12 \ln \left (\textit {\_a} \right )-12 c_{2} \right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )^{2}}d \textit {\_a} = x +c_{8} \] Verified OK. |
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\[ {}y {y^{\prime \prime }}^{3}+y^{3} y^{\prime } = 0 \] |
\[ y = \frac {{\mathrm e}^{-x}}{c_{6}} \] Verified OK. |
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\[ {}y {y^{\prime \prime }}^{3}+y^{3} {y^{\prime }}^{5} = 0 \] |
\[ \int _{}^{y}\frac {1}{-\frac {\textit {\_a}^{5}}{125}+\frac {\textit {\_a}^{2} \left (-\textit {\_a}^{2}\right )^{\frac {2}{3}} c_{1}}{25}+\frac {\textit {\_a} \left (-\textit {\_a}^{2}\right )^{\frac {1}{3}} c_{1}^{2}}{15}+\frac {c_{1}^{3}}{27}}d \textit {\_a} = x +c_{2} \] Verified OK. |
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\[ {}{y^{\prime }}^{2}+\left (b x +a y\right ) y^{\prime }+a b x y = 0 \] |
\[ y = \frac {{\mathrm e}^{-a x}}{c_{2}} \] Verified OK. |
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\[ {}{y^{\prime }}^{2}+y \left (y-x \right ) y^{\prime }-x y^{3} = 0 \] |
\[ y = c_{2} {\mathrm e}^{\frac {x^{2}}{2}} \] Verified OK. |
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\[ {}{y^{\prime }}^{2} x^{2}+3 x y y^{\prime }+2 y^{2} = 0 \] |
\[ y = \frac {c_{2}}{x^{2}} \] Verified OK. |
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\[ {}{y^{\prime }}^{2} x^{2}+4 x y y^{\prime }-5 y^{2} = 0 \] |
\[ y = c_{2} x \] Verified OK. |
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\[ {}{y^{\prime }}^{2} x^{2}+\left (x^{2} y-2 x y+x^{3}\right ) y^{\prime }+\left (y^{2}-x^{2} y\right ) \left (1-x \right ) = 0 \] |
\[ y = x \left (c_{2} -x \right ) \] Verified OK. |
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\[ {}{y^{\prime }}^{2} x^{2}+\left (a \,x^{2} y^{3}+b \right ) y^{\prime }+a b y^{3} = 0 \] |
\[ y = \frac {1}{\sqrt {2 a c_{2} +2 a x}} \] Verified OK. \[ y = -\frac {1}{\sqrt {2 a c_{2} +2 a x}} \] Verified OK. |
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\[ {}\left (-a^{2}+x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+y^{2} = 0 \] |
\[ y = \frac {c_{2}}{-x +a} \] Verified OK. |
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\[ {}y {y^{\prime }}^{2}-\left (y-x \right ) y^{\prime }-x = 0 \] |
\[ y = \sqrt {-x^{2}+2 c_{2}} \] Verified OK. \[ y = -\sqrt {-x^{2}+2 c_{2}} \] Verified OK. |
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\[ {}x y {y^{\prime }}^{2}+\left (x^{2}+y^{2}\right ) y^{\prime }+x y = 0 \] |
\[ y = \sqrt {-x^{2}+2 c_{2}} \] Verified OK. \[ y = -\sqrt {-x^{2}+2 c_{2}} \] Verified OK. |
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\[ {}x y^{2} {y^{\prime }}^{2}-2 y^{3} y^{\prime }+2 x y^{2}-x^{3} = 0 \] |
\[ \sqrt {\frac {-x^{2}+y^{2}}{x^{2}}} = c_{4} {\mathrm e}^{c_{3}} x \] Verified OK. |
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\[ {}{y^{\prime }}^{3}-\left (x^{2}+x y+y^{2}\right ) {y^{\prime }}^{2}+\left (x y^{3}+x^{2} y^{2}+x^{3} y\right ) y^{\prime }-x^{3} y^{3} = 0 \] |
\[ y = c_{3} {\mathrm e}^{\frac {x^{2}}{2}} \] Verified OK. |
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\[ {}\left (-a^{2}+x^{2}\right ) {y^{\prime }}^{3}+b x \left (-a^{2}+x^{2}\right ) {y^{\prime }}^{2}+y^{\prime }+b x = 0 \] |
\[ y = -\frac {b \,x^{2}}{2}+c_{3} \] Verified OK. |
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\[ {}{y^{\prime }}^{3} \sin \left (x \right )-\left (y \sin \left (x \right )-\cos \left (x \right )^{2}\right ) {y^{\prime }}^{2}-\left (y \cos \left (x \right )^{2}+\sin \left (x \right )\right ) y^{\prime }+y \sin \left (x \right ) = 0 \] |
\[ y = -\cos \left (x \right )+c_{2} \] Verified OK. \[ y = -\ln \left (\tan \left (\frac {x}{2}\right )\right )+c_{3} \] Verified OK. |
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\[ {}2 y {y^{\prime }}^{3}-y {y^{\prime }}^{2}+2 x y^{\prime }-x = 0 \] |
\[ y = -x \] Verified OK. \[ y = -\frac {4 x}{-2-2 i \sqrt {3}} \] Verified OK. \[ y = -\frac {4 x}{-2+2 i \sqrt {3}} \] Verified OK. \[ y = -\frac {1}{c_{3}^{2} \left (-\frac {1}{x^{\frac {3}{2}} c_{3}^{3}-1}\right )^{\frac {2}{3}}} \] Verified OK. |
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\[ {}y^{\prime \prime }-\frac {D\left (f \right )\left (y\right ) {y^{\prime }}^{3}}{f \left (y\right )}+g \left (x \right ) y^{\prime }+h \left (x \right ) f \left (y\right ) = 0 \] |
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\[ {}{y^{\prime }}^{2}+\left (x +y\right ) y^{\prime }+x y = 0 \] |
\[ y = \frac {{\mathrm e}^{-x}}{c_{2}} \] Verified OK. |
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\[ {}{y^{\prime }}^{3}-\left (y^{2}+2 x \right ) {y^{\prime }}^{2}+\left (x^{2}-y^{2}+2 x y^{2}\right ) y^{\prime }-\left (x^{2}-y^{2}\right ) y^{2} = 0 \] |
\[ y = -x -1+c_{3} {\mathrm e}^{x} \] Verified OK. |
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\[ {}{y^{\prime }}^{2} x^{2}-2 \left (x y+2 y^{\prime }\right ) y^{\prime }+y^{2} = 0 \] |
\[ y = c_{2} \left (x +2\right ) \] Verified OK. |
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\[ {}{y^{\prime }}^{2} x^{2}-\left (-1+x \right )^{2} = 0 \] |
\[ y = x -\ln \left (x \right )+c_{2} \] Verified OK. |
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\[ {}8 x^{2} y^{3}-2 y^{4}+\left (5 x^{3} y^{2}-8 x y^{3}\right ) y^{\prime } = 0 \] |
\[ \frac {5 \ln \left (y\right )}{18}+\frac {\ln \left (-x^{2}+y\right )}{6} = -\frac {\ln \left (x \right )}{9}+c_{1} \] Verified OK. |
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\[ {}x y {y^{\prime }}^{2}-\left (x^{2}+y^{2}\right ) y^{\prime }+x y = 0 \] |
\[ y = c_{2} x \] Verified OK. |
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\[ {}{y^{\prime }}^{2} = 9 y^{4} \] |
\[ y = \frac {1}{3 c_{2} +3 x} \] Verified OK. |
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\[ {}{y^{\prime }}^{3}-y^{\prime } {\mathrm e}^{2 x} = 0 \] |
\[ y = {\mathrm e}^{x}+c_{2} \] Verified OK. \[ y = -{\mathrm e}^{x}+c_{3} \] Verified OK. |
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\[ {}y = y y^{\prime }+y^{\prime }-{y^{\prime }}^{2} \] |
\[ y = c_{2} {\mathrm e}^{x} \] Verified OK. |
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\[ {}{y^{\prime }}^{2} = x^{6} \] |
\[ y = -\frac {x^{4}}{4}+c_{2} \] Verified OK. |
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\[ {}{y^{\prime }}^{2}-2 x y^{\prime }-8 x^{2} = 0 \] |
\[ y = 2 x^{2}+c_{2} \] Verified OK. |
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\[ {}{y^{\prime }}^{2} x^{2}+3 x y y^{\prime }+2 y^{2} = 0 \] |
\[ y = \frac {c_{2}}{x^{2}} \] Verified OK. |
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\[ {}{y^{\prime }}^{2}-\left (y+2 x \right ) y^{\prime }+x^{2}+x y = 0 \] |
\[ y = -1-x +c_{2} {\mathrm e}^{x} \] Verified OK. |
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\[ {}{y^{\prime }}^{3} = y {y^{\prime }}^{2}-x^{2} y^{\prime }+x^{2} y \] |
\[ y = c_{3} {\mathrm e}^{x} \] Verified OK. |
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\[ {}{y^{\prime }}^{2}-y^{2} = 0 \] |
\[ y = \frac {{\mathrm e}^{-x}}{c_{2}} \] Verified OK. |
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\[ {}\left (y^{\prime }-1\right )^{2} = y^{2} \] |
\[ y = {\mathrm e}^{x} c_{2} -1 \] Verified OK. |
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\[ {}{y^{\prime }}^{4} = 1 \] |
\[ y = i x +c_{3} \] Verified OK. \[ y = -i x +\textit {\_C4} \] Verified OK. |
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