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Mathematica |
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\[
{} y^{\prime \prime } = -\frac {a y}{\left (x^{2}-1\right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}+\frac {a^{2} y}{\left (x^{2}-1\right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (-a^{2}-\lambda \left (x^{2}-1\right )\right ) y}{\left (x^{2}-1\right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (\left (x^{2}-1\right ) \left (a \,x^{2}+b x +c \right )-k^{2}\right ) y}{\left (x^{2}-1\right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (-a^{2} \left (x^{2}-1\right )^{2}-n \left (n +1\right ) \left (x^{2}-1\right )-m^{2}\right ) y}{\left (x^{2}-1\right )^{2}}
\]
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\[
{} y^{\prime \prime } = \frac {2 x \left (2 a -1\right ) y^{\prime }}{x^{2}-1}-\frac {\left (x^{2} \left (2 a \left (2 a -1\right )-v \left (v +1\right )\right )+2 a +v \left (v +1\right )\right ) y}{\left (x^{2}-1\right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {2 x \left (n +1-2 a \right ) y^{\prime }}{x^{2}-1}-\frac {\left (4 a \,x^{2} \left (a -n \right )-\left (x^{2}-1\right ) \left (2 a +\left (v -n \right ) \left (v +n +1\right )\right )\right ) y}{\left (x^{2}-1\right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {\left (2 x^{2}+a \right ) y^{\prime }}{x \left (x^{2}+a \right )}-\frac {b y}{x^{2} \left (x^{2}+a \right )}
\]
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\[
{} y^{\prime \prime } = -\frac {b^{2} y}{\left (a^{2}+x^{2}\right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {2 \left (x^{2}-1\right ) y^{\prime }}{x \left (x -1\right )^{2}}-\frac {\left (-2 x^{2}+2 x +2\right ) y}{x^{2} \left (x -1\right )^{2}}
\]
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\[
{} y^{\prime \prime } = \frac {12 y}{\left (1+x \right )^{2} \left (x^{2}+2 x +3\right )}
\]
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\[
{} y^{\prime \prime } = -\frac {b y}{x^{2} \left (-a +x \right )^{2}}
\]
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\[
{} y^{\prime \prime } = \frac {c y}{\left (-a +x \right )^{2} \left (x -b \right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {\left (\left (\alpha +\beta +1\right ) \left (-a +x \right )^{2} \left (x -b \right )+\left (1-\alpha -\beta \right ) \left (x -b \right )^{2} \left (-a +x \right )\right ) y^{\prime }}{\left (-a +x \right )^{2} \left (x -b \right )^{2}}-\frac {\alpha \beta \left (a -b \right )^{2} y}{\left (-a +x \right )^{2} \left (x -b \right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {\left (-x^{2} \left (a^{2}-1\right )+2 \left (a +3\right ) b x -b^{2}\right ) y}{4 x^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {\left (a \,x^{2}+a -3\right ) y}{4 \left (x^{2}+1\right )^{2}}
\]
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\[
{} y^{\prime \prime } = \frac {18 y}{\left (2 x +1\right )^{2} \left (x^{2}+x +1\right )}
\]
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\[
{} y^{\prime \prime } = \frac {3 y}{4 \left (x^{2}+x +1\right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (x -1\right )}-\frac {\left (v \left (v +1\right ) \left (x -1\right )-a^{2} x \right ) y}{4 x^{2} \left (x -1\right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (x -1\right )}-\frac {\left (-v \left (v +1\right ) \left (x -1\right )^{2}-4 n^{2} x \right ) y}{4 x^{2} \left (x -1\right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {3 y}{16 x^{2} \left (x -1\right )^{2}}
\]
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\[
{} y^{\prime \prime } = \frac {\left (7 a \,x^{2}+5\right ) y^{\prime }}{x \left (a \,x^{2}+1\right )}-\frac {\left (15 a \,x^{2}+5\right ) y}{x^{2} \left (a \,x^{2}+1\right )}
\]
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\[
{} y^{\prime \prime } = -\frac {b x y^{\prime }}{\left (x^{2}-1\right ) a}-\frac {\left (c \,x^{2}+d x +e \right ) y}{a \left (x^{2}-1\right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {\left (b \,x^{2}+c x +d \right ) y}{a \,x^{2} \left (x -1\right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {2 y^{\prime }}{x}-\frac {c y}{x^{2} \left (a x +b \right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {y}{\left (a x +b \right )^{4}}
\]
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\[
{} y^{\prime \prime } = -\frac {A y}{\left (a \,x^{2}+b x +c \right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {y^{\prime }}{x^{4}}+\frac {y}{x^{5}}
\]
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\[
{} y^{\prime \prime } = -\frac {\left (3 x^{2}-1\right ) y^{\prime }}{\left (x^{2}-1\right ) x}-\frac {\left (x^{2}-1-\left (2 v +1\right )^{2}\right ) y}{\left (x^{2}-1\right )^{2}}
\]
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\[
{} y^{\prime \prime } = \frac {\left (3 x +1\right ) y^{\prime }}{\left (x -1\right ) \left (1+x \right )}-\frac {36 \left (1+x \right )^{2} y}{\left (x -1\right )^{2} \left (3 x +5\right )^{2}}
\]
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\[
{} y^{\prime \prime } = \frac {y^{\prime }}{x}-\frac {a y}{x^{6}}
\]
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\[
{} y^{\prime \prime } = -\frac {\left (3 x^{2}+a \right ) y^{\prime }}{x^{3}}-\frac {b y}{x^{6}}
\]
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\[
{} y^{\prime \prime } = -\frac {\left (\left (1-4 a \right ) x^{2}-1\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (\left (-v^{2}+x^{2}\right ) \left (x^{2}-1\right )^{2}+4 a \left (a +1\right ) x^{4}-2 a \,x^{2} \left (x^{2}-1\right )\right ) y}{x^{2} \left (x^{2}-1\right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\left (\frac {1-\operatorname {a1} -\operatorname {b1}}{x -\operatorname {c1}}+\frac {1-\operatorname {a2} -\operatorname {b2}}{x -\operatorname {c2}}+\frac {1-\operatorname {a3} -\operatorname {b3}}{x -\operatorname {c3}}\right ) y^{\prime }-\frac {\left (\frac {\operatorname {a1} \operatorname {b1} \left (\operatorname {c1} -\operatorname {c3} \right ) \left (\operatorname {c1} -\operatorname {c2} \right )}{x -\operatorname {c1}}+\frac {\operatorname {a2} \operatorname {b2} \left (\operatorname {c2} -\operatorname {c1} \right ) \left (\operatorname {c2} -\operatorname {c3} \right )}{x -\operatorname {c2}}+\frac {\operatorname {a3} \operatorname {b3} \left (\operatorname {c3} -\operatorname {c2} \right ) \left (\operatorname {c3} -\operatorname {c1} \right )}{x -\operatorname {c3}}\right ) y}{\left (x -\operatorname {c1} \right ) \left (x -\operatorname {c2} \right ) \left (x -\operatorname {c3} \right )}
\]
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\[
{} y^{\prime \prime } = -\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {\left (-2 x^{2}+1\right ) y}{4 x^{6}}
\]
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\[
{} y^{\prime \prime } = \frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {\left (a \,x^{4}+10 x^{2}+1\right ) y}{4 x^{6}}
\]
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\[
{} y^{\prime \prime } = -\frac {27 x y}{16 \left (x^{3}-1\right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\left (\frac {\left (1-\operatorname {al1} -\operatorname {bl1} \right ) \operatorname {b1}}{\operatorname {b1} x -\operatorname {a1}}+\frac {\left (1-\operatorname {al2} -\operatorname {bl2} \right ) \operatorname {b2}}{\operatorname {b2} x -\operatorname {a2}}+\frac {\left (1-\operatorname {al3} -\operatorname {bl3} \right ) \operatorname {b3}}{\operatorname {b3} x -\operatorname {a3}}\right ) y^{\prime }-\frac {\left (\frac {\operatorname {al1} \operatorname {bl1} \left (\operatorname {a1} \operatorname {b2} -\operatorname {b1} \operatorname {a2} \right ) \left (-\operatorname {a1} \operatorname {b3} +\operatorname {a3} \operatorname {b1} \right )}{\operatorname {b1} x -\operatorname {a1}}+\frac {\operatorname {al2} \operatorname {bl2} \left (\operatorname {a2} \operatorname {b3} -\operatorname {a3} \operatorname {b2} \right ) \left (\operatorname {a1} \operatorname {b2} -\operatorname {b1} \operatorname {a2} \right )}{\operatorname {b2} x -\operatorname {a2}}+\frac {\operatorname {al3} \operatorname {bl3} \left (-\operatorname {a1} \operatorname {b3} +\operatorname {a3} \operatorname {b1} \right ) \left (\operatorname {a2} \operatorname {b3} -\operatorname {a3} \operatorname {b2} \right )}{\operatorname {b3} x -\operatorname {a3}}\right ) y}{\left (\operatorname {b1} x -\operatorname {a1} \right ) \left (\operatorname {b2} x -\operatorname {a2} \right ) \left (\operatorname {b3} x -\operatorname {a3} \right )}
\]
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\[
{} y^{\prime \prime } = -\frac {\left (x^{2} \left (\left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right )+\left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )+\left (x^{2}-\operatorname {a3} \right ) \left (x^{2}-\operatorname {a1} \right )\right )-\left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )\right ) y^{\prime }}{x \left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )}-\frac {\left (A \,x^{2}+B \right ) y}{x \left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )}
\]
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\[
{} y^{\prime \prime } = -a \,x^{2 a -1} x^{-2 a} y^{\prime }-b^{2} x^{-2 a} y
\]
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\[
{} y^{\prime \prime } = -\frac {\left (a p \,x^{b}+q \right ) y^{\prime }}{x \left (a \,x^{b}-1\right )}-\frac {\left (a r \,x^{b}+s \right ) y}{x^{2} \left (a \,x^{b}-1\right )}
\]
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\[
{} y^{\prime \prime } = \frac {y}{1+{\mathrm e}^{x}}
\]
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\[
{} y^{\prime \prime } = \frac {y^{\prime }}{x \ln \left (x \right )}+\ln \left (x \right )^{2} y
\]
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\[
{} y^{\prime \prime } = \frac {y^{\prime }}{x \left (-1+\ln \left (x \right )\right )}-\frac {y}{x^{2} \left (-1+\ln \left (x \right )\right )}
\]
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\[
{} y^{\prime \prime } = -\frac {\left (-a^{2} \sinh \left (x \right )^{2}-n \left (n -1\right )\right ) y}{\sinh \left (x \right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {2 n \cosh \left (x \right ) y^{\prime }}{\sinh \left (x \right )}-\left (-a^{2}+n^{2}\right ) y
\]
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\[
{} y^{\prime \prime } = -\frac {\left (2 n +1\right ) \cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}-\left (v +n +1\right ) \left (v -n \right ) y
\]
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\[
{} y^{\prime \prime } = -\frac {\left (\sin \left (x \right )^{2}-\cos \left (x \right )\right ) y^{\prime }}{\sin \left (x \right )}-y \sin \left (x \right )^{2}
\]
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\[
{} y^{\prime \prime } = -\frac {x \sin \left (x \right ) y^{\prime }}{x \cos \left (x \right )-\sin \left (x \right )}+\frac {\sin \left (x \right ) y}{x \cos \left (x \right )-\sin \left (x \right )}
\]
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\[
{} y^{\prime \prime } = -\frac {\left (x^{2} \sin \left (x \right )-2 x \cos \left (x \right )\right ) y^{\prime }}{x^{2} \cos \left (x \right )}-\frac {\left (2 \cos \left (x \right )-x \sin \left (x \right )\right ) y}{x^{2} \cos \left (x \right )}
\]
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\[
{} \cos \left (x \right )^{2} y^{\prime \prime }-\left (a \cos \left (x \right )^{2}+n \left (n -1\right )\right ) y = 0
\]
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\[
{} y^{\prime \prime } = -\frac {a \left (n -1\right ) \sin \left (2 a x \right ) y^{\prime }}{\cos \left (a x \right )^{2}}-\frac {n \,a^{2} \left (\left (n -1\right ) \sin \left (a x \right )^{2}+\cos \left (a x \right )^{2}\right ) y}{\cos \left (a x \right )^{2}}
\]
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\[
{} y^{\prime \prime } = \frac {2 y}{\sin \left (x \right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {a y}{\sin \left (x \right )^{2}}
\]
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\[
{} \sin \left (x \right )^{2} y^{\prime \prime }-\left (a \sin \left (x \right )^{2}+n \left (n -1\right )\right ) y = 0
\]
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\[
{} y^{\prime \prime } = -\frac {\left (-a^{2} \cos \left (x \right )^{2}-\left (3-2 a \right ) \cos \left (x \right )-3+3 a \right ) y}{\sin \left (x \right )^{2}}
\]
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\[
{} \sin \left (x \right )^{2} y^{\prime \prime }-\left (a^{2} \cos \left (x \right )^{2}+b \cos \left (x \right )+\frac {b^{2}}{\left (2 a -3\right )^{2}}+3 a +2\right ) y = 0
\]
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\[
{} y^{\prime \prime } = -\frac {\left (-\left (a^{2} b^{2}-\left (a +1\right )^{2}\right ) \sin \left (x \right )^{2}-a \left (a +1\right ) b \sin \left (2 x \right )-a \left (a -1\right )\right ) y}{\sin \left (x \right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {\left (a \cos \left (x \right )^{2}+b \sin \left (x \right )^{2}+c \right ) y}{\sin \left (x \right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {\cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}+\frac {y}{\sin \left (x \right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {\cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}-\frac {\left (v \left (v +1\right ) \sin \left (x \right )^{2}-n^{2}\right ) y}{\sin \left (x \right )^{2}}
\]
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\[
{} y^{\prime \prime } = \frac {\cos \left (2 x \right ) y^{\prime }}{\sin \left (2 x \right )}-2 y
\]
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\[
{} y^{\prime \prime } = -\frac {\cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}-\frac {\left (-17 \sin \left (x \right )^{2}-1\right ) y}{4 \sin \left (x \right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {b \cos \left (x \right ) y^{\prime }}{\sin \left (x \right ) a}-\frac {\left (c \cos \left (x \right )^{2}+d \cos \left (x \right )+e \right ) y}{a \sin \left (x \right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {4 \sin \left (3 x \right ) y}{\sin \left (x \right )^{3}}
\]
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\[
{} y^{\prime \prime } = -\frac {\left (4 v \left (v +1\right ) \sin \left (x \right )^{2}-\cos \left (x \right )^{2}+2-4 n^{2}\right ) y}{4 \sin \left (x \right )^{2}}
\]
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\[
{} y^{\prime \prime } = \frac {\left (3 \sin \left (x \right )^{2}+1\right ) y^{\prime }}{\cos \left (x \right ) \sin \left (x \right )}+\frac {\sin \left (x \right )^{2} y}{\cos \left (x \right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {\left (-a \cos \left (x \right )^{2} \sin \left (x \right )^{2}-m \left (m -1\right ) \sin \left (x \right )^{2}-n \left (n -1\right ) \cos \left (x \right )^{2}\right ) y}{\cos \left (x \right )^{2} \sin \left (x \right )^{2}}
\]
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\[
{} y^{\prime \prime } = -\frac {x y^{\prime }}{f \left (x \right )}+\frac {y}{f \left (x \right )}
\]
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\[
{} y^{\prime \prime } = -\frac {f^{\prime }\left (x \right ) y^{\prime }}{2 f \left (x \right )}-\frac {g \left (x \right ) y}{f \left (x \right )}
\]
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\[
{} y^{\prime \prime } = -\frac {\left (2 f \left (x \right ) {g^{\prime }\left (x \right )}^{2} g \left (x \right )-\left (g \left (x \right )^{2}-1\right ) \left (f \left (x \right ) g^{\prime \prime }\left (x \right )+2 f^{\prime }\left (x \right ) g^{\prime }\left (x \right )\right )\right ) y^{\prime }}{f \left (x \right ) g^{\prime }\left (x \right ) \left (g \left (x \right )^{2}-1\right )}-\frac {\left (\left (g \left (x \right )^{2}-1\right ) \left (f^{\prime }\left (x \right ) \left (f \left (x \right ) g^{\prime \prime }\left (x \right )+2 f^{\prime }\left (x \right ) g^{\prime }\left (x \right )\right )-f \left (x \right ) f^{\prime \prime }\left (x \right ) g^{\prime }\left (x \right )\right )-\left (2 f^{\prime }\left (x \right ) g \left (x \right )+v \left (v +1\right ) f \left (x \right ) g^{\prime }\left (x \right )\right ) f \left (x \right ) {g^{\prime }\left (x \right )}^{2}\right ) y}{f \left (x \right )^{2} g^{\prime }\left (x \right ) \left (g \left (x \right )^{2}-1\right )}
\]
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\[
{} y^{\prime \prime } = -\frac {y^{\prime }}{x}-\frac {\left (x -1\right ) y}{x^{4}}
\]
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\[
{} y^{\prime \prime } = -\frac {y^{\prime }}{x}-\frac {\left (-x -1\right ) y}{x^{4}}
\]
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\[
{} y^{\prime \prime } = -\frac {b^{2} y}{\left (-a^{2}+x^{2}\right )^{2}}
\]
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\[
{} y^{\prime \prime }-y^{2} = 0
\]
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\[
{} y^{\prime \prime }-6 y^{2} = 0
\]
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\[
{} y^{\prime \prime }-6 y^{2}+4 y = 0
\]
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\[
{} y^{\prime \prime }-a y^{3} = 0
\]
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\[
{} y^{\prime \prime }+a \,x^{r} y^{2} = 0
\]
|
✗ |
✗ |
✗ |
|
|
\[
{} y^{\prime \prime }+6 a^{10} y^{11}-y = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime \prime }-\frac {1}{\left (y^{2} a +b x y+c \,x^{2}+\alpha y+\beta x +\gamma \right )^{{3}/{2}}} = 0
\]
|
✗ |
✓ |
✗ |
|
|
\[
{} y^{\prime \prime }-{\mathrm e}^{y} = 0
\]
|
✓ |
✓ |
✓ |
|
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\[
{} y^{\prime \prime }+a \,{\mathrm e}^{x} \sqrt {y} = 0
\]
|
✗ |
✗ |
✗ |
|
|
\[
{} y^{\prime \prime }+{\mathrm e}^{x} \sin \left (y\right ) = 0
\]
|
✗ |
✗ |
✗ |
|
|
\[
{} y^{\prime \prime }+a \sin \left (y\right ) = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime \prime } = \frac {f \left (\frac {y}{\sqrt {x}}\right )}{x^{{3}/{2}}}
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime \prime }-3 y^{\prime }-y^{2}-2 y = 0
\]
|
✗ |
✗ |
✗ |
|
|
\[
{} y^{\prime \prime }-7 y^{\prime }-y^{{3}/{2}}+12 y = 0
\]
|
✗ |
✗ |
✗ |
|
|
\[
{} y^{\prime \prime }+5 a y^{\prime }-6 y^{2}+6 a^{2} y = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime \prime }+3 a y^{\prime }-2 y^{3}+2 a^{2} y = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime \prime }-\frac {\left (3 n +4\right ) y^{\prime }}{n}-\frac {2 \left (n +1\right ) \left (n +2\right ) y \left (y^{\frac {n}{n +1}}-1\right )}{n^{2}} = 0
\]
|
✗ |
✗ |
✗ |
|
|
\[
{} y^{\prime \prime }+a y^{\prime }+b y^{n}+\frac {\left (a^{2}-1\right ) y}{4} = 0
\]
|
✗ |
✗ |
✗ |
|
|
\[
{} y^{\prime \prime }+a y^{\prime }+b \,x^{v} y^{n} = 0
\]
|
✗ |
✗ |
✗ |
|
|
\[
{} y^{\prime \prime }+a y^{\prime }+f \left (x \right ) \sin \left (y\right ) = 0
\]
|
✗ |
✗ |
✗ |
|
|
\[
{} y^{\prime \prime }+y y^{\prime }-y^{3} = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime \prime }+y y^{\prime }-y^{3}+a y = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime \prime }+\left (y+3 a \right ) y^{\prime }-y^{3}+y^{2} a +2 a^{2} y = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime \prime }+\left (y+3 f \left (x \right )\right ) y^{\prime }-y^{3}+y^{2} f \left (x \right )+y \left (f^{\prime }\left (x \right )+2 f \left (x \right )^{2}\right ) = 0
\]
|
✗ |
✗ |
✗ |
|
|
\[
{} y^{\prime \prime }+\left (3 y+f \left (x \right )\right ) y^{\prime }+y^{3}+y^{2} f \left (x \right ) = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} y^{\prime \prime }-\left (3 y+f \left (x \right )\right ) y^{\prime }+y^{3}+y^{2} f \left (x \right ) = 0
\]
|
✓ |
✓ |
✗ |
|