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