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# |
ODE |
Mathematica |
Maple |
Sympy |
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\[
{} x^{2} y^{\prime \prime }-2 a x y^{\prime }+\left (-b^{2} x^{2}+a \left (a +1\right )\right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} x^{2} y^{\prime \prime }+\lambda x y^{\prime }+\left (a \,x^{2}+b x +c \right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} x^{2} y^{\prime \prime }+a x y^{\prime }+\left (b \,x^{n}+c \right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} x^{2} y^{\prime \prime }+a x y^{\prime }+x^{n} \left (b \,x^{n}+c \right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} x^{2} y^{\prime \prime }+a \,x^{2} y^{\prime }+\left (b \,x^{2}+c x +d \right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} x^{2} y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c \left (\left (-c +a \right ) x^{2}+b \right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }-b y = 0
\]
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✓ |
✓ |
✓ |
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\[
{} x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (k \left (a -k \right ) x^{2}+\left (a n +b k -2 k n \right ) x +n \left (b -n -1\right )\right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} a_{2} x^{2} y^{\prime \prime }+\left (a_{1} x^{2}+b_{1} x \right ) y^{\prime }+\left (a_{0} x^{2}+b_{0} x +c_{0} \right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} x^{2} y^{\prime \prime }+\left (a \,x^{2}+\left (a b -1\right ) x +b \right ) y^{\prime }+a^{2} b x y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} x^{2} y^{\prime \prime }-2 x \left (x^{2}-a \right ) y^{\prime }+\left (2 n \,x^{2}+\left (\left (-1\right )^{n}-1\right ) a \right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} x^{2} y^{\prime \prime }+x \left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (A \,x^{3}+B \,x^{2}+C x +d \right ) y = 0
\]
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✗ |
✓ |
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\[
{} x^{2} y^{\prime \prime }+a \,x^{n} y^{\prime }-\left (a b \,x^{n}+a c \,x^{n -1}+b^{2} x^{2}+2 b x c +c^{2}-c \right ) y = 0
\]
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✗ |
✗ |
✗ |
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\[
{} x^{2} y^{\prime \prime }+a \,x^{n} y^{\prime }+\left (a b \,x^{n +2 m}-b^{2} x^{4 m +2}+a m \,x^{n -1}-m^{2}-m \right ) y = 0
\]
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✗ |
✗ |
✗ |
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\[
{} x^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime } x +b \left (a \,x^{n}-1\right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} x^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime } x +\left (\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} x^{2} y^{\prime \prime }+x \left (2 a \,x^{n}+b \right ) y^{\prime }+\left (a^{2} x^{2 n}+a \left (b +n -1\right ) x^{n}+\alpha \,x^{2 m}+\beta \,x^{m}+\gamma \right ) y = 0
\]
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✓ |
✓ |
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\[
{} x^{2} y^{\prime \prime }+\left (a \,x^{n +2}+b \,x^{2}+c \right ) y^{\prime }+\left (a n \,x^{n +1}+a c \,x^{n}+b c \right ) y = 0
\]
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✗ |
✗ |
✗ |
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\[
{} \left (-x^{2}+1\right ) y^{\prime \prime }+n \left (n -1\right ) y = 0
\]
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✓ |
✓ |
✓ |
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\[
{} \left (-a^{2}+x^{2}\right ) y^{\prime \prime }+b y^{\prime }-6 y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (x^{2}-1\right ) y^{\prime \prime }+x y^{\prime }+a y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+n^{2} y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+n \left (n +1\right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+\nu \left (\nu +1\right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (-x^{2}+1\right ) y^{\prime \prime }-3 x y^{\prime }+n \left (n +2\right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (x^{2}-1\right ) y^{\prime \prime }+2 \left (n +1\right ) x y^{\prime }-\left (\nu +n +1\right ) \left (\nu -n \right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (x^{2}-1\right ) y^{\prime \prime }-2 \left (n -1\right ) x y^{\prime }-\left (\nu -n +1\right ) \left (\nu +n \right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (x^{2}-1\right ) y^{\prime \prime }+\left (2 a +1\right ) y^{\prime }-b \left (2 a +b \right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (-x^{2}+1\right ) y^{\prime \prime }+\left (2 a -3\right ) x y^{\prime }+\left (n +1\right ) \left (n +2 a -1\right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (-x^{2}+1\right ) y^{\prime \prime }+\left (\beta -\alpha -\left (\alpha +\beta +2\right ) x \right ) y^{\prime }+n \left (n +\alpha +\beta +1\right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (-x^{2}+1\right ) y^{\prime \prime }+\left (\alpha -\beta +\left (\alpha +\beta -2\right ) x \right ) y^{\prime }+\left (n +1\right ) \left (n +\alpha +\beta \right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (a \,x^{2}+b \right ) y^{\prime \prime }+a x y^{\prime }+c y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (x^{2}+a \right ) y^{\prime \prime }+2 b x y^{\prime }+2 \left (b -1\right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (-a^{2}+x^{2}\right ) y^{\prime \prime }+2 b x y^{\prime }+b \left (b -1\right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (a^{2}+x^{2}\right ) y^{\prime \prime }+2 b x y^{\prime }+b \left (b -1\right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (2 n +1\right ) a x y^{\prime }+c y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+\left (2 a \,x^{2}+b \right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (-x^{2}+1\right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (c \,x^{2}+d \right ) y^{\prime }+\lambda \left (\left (-a \lambda +c \right ) x^{2}+d -b \lambda \right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (\lambda \left (a +c \right ) x^{2}+\left (c -a \right ) x +2 b \lambda \right ) y^{\prime }+\lambda ^{2} \left (c \,x^{2}+b \right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} x \left (x -1\right ) y^{\prime \prime }+\left (\left (\alpha +\beta +1\right ) x -\gamma \right ) y^{\prime }+\alpha \beta y = 0
\]
|
✓ |
✓ |
✗ |
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\[
{} x \left (x +a \right ) y^{\prime \prime }+\left (b x +c \right ) y^{\prime }+d y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} 2 x \left (x -1\right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }+\left (a x +b \right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (2 a x +x^{2}+b \right ) y^{\prime \prime }+\left (x +a \right ) y^{\prime }-m^{2} y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (d x +k \right ) y^{\prime }+\left (d -2 a \right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (k x +d \right ) y^{\prime }-k y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (a \,x^{2}+2 b x +c \right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+d y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (a \,x^{2}+2 b x +c \right ) y^{\prime \prime }+3 \left (a x +b \right ) y^{\prime }+d y = 0
\]
|
✓ |
✓ |
✗ |
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\[
{} \left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime \prime }+\left (b_{1} x +c_{1} \right ) y^{\prime }+c_{0} y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} \left (a \,x^{2}+b x +c \right ) y^{\prime \prime }-\left (-k^{2}+x^{2}\right ) y^{\prime }+\left (x +k \right ) y = 0
\]
|
✓ |
✓ |
✗ |
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\[
{} \left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (k^{3}+x^{3}\right ) y^{\prime }-\left (k^{2}-k x +x^{2}\right ) y = 0
\]
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✓ |
✓ |
✗ |
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\[
{} x^{3} y^{\prime \prime }+\left (a x +b \right ) y = 0
\]
|
✓ |
✓ |
✓ |
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\[
{} x^{3} y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c x y = 0
\]
|
✓ |
✓ |
✗ |
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\[
{} x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+b y = 0
\]
|
✓ |
✓ |
✗ |
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\[
{} x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+c y = 0
\]
|
✓ |
✓ |
✗ |
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\[
{} x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (c x +d \right ) y = 0
\]
|
✓ |
✓ |
✗ |
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\[
{} x^{3} y^{\prime \prime }+\left (a \,x^{3}+a b x -x^{2}+b \right ) y^{\prime }+a^{2} b x y = 0
\]
|
✓ |
✓ |
✗ |
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\[
{} x^{3} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime } x -\left (a \,x^{n}-a b \,x^{n -1}+b \right ) y = 0
\]
|
✗ |
✗ |
✗ |
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\[
{} x \left (a \,x^{2}+b \right ) y^{\prime \prime }+2 \left (a \,x^{2}+b \right ) y^{\prime }-2 a x y = 0
\]
|
✓ |
✓ |
✗ |
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\[
{} x \left (x^{2}+a \right ) y^{\prime \prime }+\left (b \,x^{2}+c \right ) y^{\prime }+s x y = 0
\]
|
✓ |
✓ |
✗ |
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\[
{} x^{2} \left (a x +b \right ) y^{\prime \prime }+\left (c \,x^{2}+\left (a \lambda +2 b \right ) x +b \lambda \right ) y^{\prime }+\lambda \left (c -2 a \right ) y = 0
\]
|
✓ |
✓ |
✗ |
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\[
{} x^{2} \left (a x +b \right ) y^{\prime \prime }-2 x \left (a x +2 b \right ) y^{\prime }+2 \left (a x +3 b \right ) y = 0
\]
|
✓ |
✓ |
✗ |
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\[
{} x^{2} \left (a x +b \right ) y^{\prime \prime }+\left (a \left (2-n -m \right ) x^{2}-b \left (n +m \right ) x \right ) y^{\prime }+\left (a m \left (n -1\right ) x +b n \left (m +1\right )\right ) y = 0
\]
|
✓ |
✓ |
✗ |
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\[
{} x^{2} \left (x +a_{2} \right ) y^{\prime \prime }+x \left (b_{1} x +a_{1} \right ) y^{\prime }+\left (b_{0} x +a_{0} \right ) y = 0
\]
|
✓ |
✓ |
✗ |
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\[
{} \left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +2 c \right ) y^{\prime }+\left (\beta -2 b \right ) y = 0
\]
|
✓ |
✓ |
✗ |
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\[
{} \left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +2 c \right ) y^{\prime }-\left (\alpha x +2 b -\beta \right ) y = 0
\]
|
✓ |
✓ |
✗ |
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\[
{} \left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (-2 a \,x^{2}-\left (b +1\right ) x +k \right ) y^{\prime }+2 \left (a x +1\right ) y = 0
\]
|
✓ |
✓ |
✗ |
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\[
{} \left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (n \,x^{2}+m x +k \right ) y^{\prime }+\left (k -1\right ) \left (\left (-a k +n \right ) x +m -b k \right ) y = 0
\]
|
✓ |
✗ |
✗ |
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\[
{} \left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\left (m -a \right ) x^{2}+\left (2 c m -1\right ) x -c \right ) y^{\prime }+\left (-2 m x +1\right ) y = 0
\]
|
✓ |
✗ |
✗ |
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\[
{} \left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (n \,x^{2}+m x +k \right ) y^{\prime }+\left (-2 \left (a +n \right ) x +1\right ) y = 0
\]
|
✓ |
✗ |
✗ |
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\[
{} \left (a \,x^{3}+x^{2}+b \right ) y^{\prime \prime }+a^{2} x \left (x^{2}-b \right ) y^{\prime }-a^{3} b x y = 0
\]
|
✗ |
✓ |
✗ |
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\[
{} 2 x \left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (a \,x^{2}-c \right ) y^{\prime }+\lambda \,x^{2} y = 0
\]
|
✓ |
✓ |
✗ |
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\[
{} x \left (a \,x^{2}+b x +1\right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y^{\prime }+\left (n x +m \right ) y = 0
\]
|
✓ |
✗ |
✗ |
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\[
{} x \left (x -1\right ) \left (-a +x \right ) y^{\prime \prime }+\left (\left (\alpha +\beta +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +d \right )-a \right ) x +a \gamma \right ) y^{\prime }+\left (\alpha \beta x -q \right ) y = 0
\]
|
✓ |
✓ |
✗ |
|
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\[
{} \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }-\left (-\lambda ^{2}+x^{2}\right ) y^{\prime }+\left (x +\lambda \right ) y = 0
\]
|
✗ |
✓ |
✗ |
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\[
{} 2 \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (3 a \,x^{2}+2 b x +c \right ) y^{\prime }+\lambda y = 0
\]
|
✗ |
✓ |
✗ |
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\[
{} 2 \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+3 \left (3 a \,x^{2}+2 b x +c \right ) y^{\prime }+\left (6 a x +2 b +\lambda \right ) y = 0
\]
|
✓ |
✓ |
✗ |
|
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\[
{} \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\left (\alpha \gamma +\beta \right ) x +\beta \lambda \right ) y^{\prime }-\left (\alpha x +\beta \right ) y = 0
\]
|
✗ |
✗ |
✗ |
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\[
{} \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (\lambda ^{3}+x^{3}\right ) y^{\prime }-\left (\lambda ^{2}-\lambda x +x^{2}\right ) y = 0
\]
|
✓ |
✓ |
✗ |
|
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\[
{} 2 x \left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (a \left (2-k \right ) x^{2}+b \left (1-k \right ) x -c k \right ) y^{\prime }+\lambda \,x^{k +1} y = 0
\]
|
✓ |
✗ |
✗ |
|
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\[
{} x^{4} y^{\prime \prime }+a y = 0
\]
|
✓ |
✓ |
✓ |
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\[
{} x^{4} y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y = 0
\]
|
✗ |
✓ |
✗ |
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\[
{} x^{4} y^{\prime \prime }-\left (a +b \right ) x^{2} y^{\prime }+\left (\left (a +b \right ) x +a b \right ) y = 0
\]
|
✓ |
✓ |
✗ |
|
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\[
{} x^{4} y^{\prime \prime }+2 x^{2} \left (x +a \right ) y^{\prime }+b y = 0
\]
|
✓ |
✓ |
✗ |
|
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\[
{} x^{4} y^{\prime \prime }+a \,x^{n} y^{\prime }-\left (a \,x^{n -1}+a b \,x^{n -2}+b^{2}\right ) y = 0
\]
|
✗ |
✗ |
✗ |
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\[
{} x^{2} \left (-a +x \right )^{2} y^{\prime \prime }+b y = 0
\]
|
✓ |
✓ |
✗ |
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\[
{} a \,x^{2} \left (x -1\right )^{2} y^{\prime \prime }+\left (b \,x^{2}+c x +d \right ) y = 0
\]
|
✓ |
✓ |
✗ |
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\[
{} x^{2} \left (x^{2}+a \right ) y^{\prime \prime }+\left (b \,x^{2}+c \right ) x y^{\prime }+d y = 0
\]
|
✓ |
✓ |
✗ |
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\[
{} \left (x^{2}+1\right )^{2} y^{\prime \prime }+a y = 0
\]
|
✓ |
✓ |
✓ |
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\[
{} \left (x^{2}-1\right )^{2} y^{\prime \prime }+a y = 0
\]
|
✓ |
✓ |
✓ |
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\[
{} \left (a^{2}+x^{2}\right )^{2} y^{\prime \prime }+b^{2} y = 0
\]
|
✓ |
✓ |
✓ |
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\[
{} \left (-a^{2}+x^{2}\right )^{2} y^{\prime \prime }+b^{2} y = 0
\]
|
✓ |
✓ |
✓ |
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\[
{} 4 \left (x^{2}+1\right )^{2} y^{\prime \prime }+\left (a \,x^{2}+a -3\right ) y = 0
\]
|
✓ |
✓ |
✓ |
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\[
{} \left (a \,x^{2}+b \right )^{2} y^{\prime \prime }+2 a x \left (a \,x^{2}+b \right ) y^{\prime }+c y = 0
\]
|
✓ |
✓ |
✗ |
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\[
{} \left (x^{2}-1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}-1\right ) y^{\prime }-\left (\nu \left (\nu +1\right ) \left (x^{2}-1\right )+n^{2}\right ) y = 0
\]
|
✓ |
✓ |
✗ |
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\[
{} \left (-x^{2}+1\right )^{2} y^{\prime \prime }-2 x \left (-x^{2}+1\right ) y^{\prime }+\left (\nu \left (\nu +1\right ) \left (-x^{2}+1\right )-\mu ^{2}\right ) y = 0
\]
|
✓ |
✓ |
✗ |
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|
\[
{} a \left (x^{2}-1\right )^{2} y^{\prime \prime }+b x \left (x^{2}-1\right ) y^{\prime }+\left (c \,x^{2}+d x +e \right ) y = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} \left (a \,x^{2}+b \right )^{2} y^{\prime \prime }+\left (2 a x +c \right ) \left (a \,x^{2}+b \right ) y^{\prime }+k y = 0
\]
|
✓ |
✓ |
✗ |
|
|
\[
{} \left (a \,x^{2}+b \right )^{2} y^{\prime \prime }+\left (a \,x^{2}+b \right ) \left (c \,x^{2}+d \right ) y^{\prime }+2 \left (-a d +b c \right ) x y = 0
\]
|
✓ |
✓ |
✓ |
|