4.15.32 \(\left (x^2+1\right )^{3/2} \left (y(x)+\sqrt {y(x)^2+1}\right ) y'(x)=y(x)^2+1\)

ODE
\[ \left (x^2+1\right )^{3/2} \left (y(x)+\sqrt {y(x)^2+1}\right ) y'(x)=y(x)^2+1 \] ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.108219 (sec), leaf count = 321

\[\left \{\left \{y(x)\to -\frac {i \left (\sinh \left (c_1+\frac {x}{\sqrt {x^2+1}}\right )+\cosh \left (c_1+\frac {x}{\sqrt {x^2+1}}\right )-1\right )}{\sqrt {-2 \sinh \left (c_1+\frac {x}{\sqrt {x^2+1}}\right )-2 \cosh \left (c_1+\frac {x}{\sqrt {x^2+1}}\right )+1}}\right \},\left \{y(x)\to \frac {i \left (\sinh \left (c_1+\frac {x}{\sqrt {x^2+1}}\right )+\cosh \left (c_1+\frac {x}{\sqrt {x^2+1}}\right )-1\right )}{\sqrt {-2 \sinh \left (c_1+\frac {x}{\sqrt {x^2+1}}\right )-2 \cosh \left (c_1+\frac {x}{\sqrt {x^2+1}}\right )+1}}\right \},\left \{y(x)\to -\frac {i \left (\sinh \left (c_1+\frac {x}{\sqrt {x^2+1}}\right )+\cosh \left (c_1+\frac {x}{\sqrt {x^2+1}}\right )+1\right )}{\sqrt {2 \sinh \left (c_1+\frac {x}{\sqrt {x^2+1}}\right )+2 \cosh \left (c_1+\frac {x}{\sqrt {x^2+1}}\right )+1}}\right \},\left \{y(x)\to \frac {i \left (\sinh \left (c_1+\frac {x}{\sqrt {x^2+1}}\right )+\cosh \left (c_1+\frac {x}{\sqrt {x^2+1}}\right )+1\right )}{\sqrt {2 \sinh \left (c_1+\frac {x}{\sqrt {x^2+1}}\right )+2 \cosh \left (c_1+\frac {x}{\sqrt {x^2+1}}\right )+1}}\right \}\right \}\]

Maple
cpu = 0.01 (sec), leaf count = 28

\[ \left \{ {x{\frac {1}{\sqrt {{x}^{2}+1}}}}-{\it Arcsinh} \left (y \relax (x ) \right ) -{\frac {\ln \left (1+ \left (y \relax (x ) \right ) ^{2} \right ) }{2}}+{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[(1 + x^2)^(3/2)*(y[x] + Sqrt[1 + y[x]^2])*y'[x] == 1 + y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> ((-I)*(-1 + Cosh[x/Sqrt[1 + x^2] + C[1]] + Sinh[x/Sqrt[1 + x^2] + C[1]
]))/Sqrt[1 - 2*Cosh[x/Sqrt[1 + x^2] + C[1]] - 2*Sinh[x/Sqrt[1 + x^2] + C[1]]]}, 
{y[x] -> (I*(-1 + Cosh[x/Sqrt[1 + x^2] + C[1]] + Sinh[x/Sqrt[1 + x^2] + C[1]]))/
Sqrt[1 - 2*Cosh[x/Sqrt[1 + x^2] + C[1]] - 2*Sinh[x/Sqrt[1 + x^2] + C[1]]]}, {y[x
] -> ((-I)*(1 + Cosh[x/Sqrt[1 + x^2] + C[1]] + Sinh[x/Sqrt[1 + x^2] + C[1]]))/Sq
rt[1 + 2*Cosh[x/Sqrt[1 + x^2] + C[1]] + 2*Sinh[x/Sqrt[1 + x^2] + C[1]]]}, {y[x] 
-> (I*(1 + Cosh[x/Sqrt[1 + x^2] + C[1]] + Sinh[x/Sqrt[1 + x^2] + C[1]]))/Sqrt[1 
+ 2*Cosh[x/Sqrt[1 + x^2] + C[1]] + 2*Sinh[x/Sqrt[1 + x^2] + C[1]]]}}

Maple raw input

dsolve((y(x)+(1+y(x)^2)^(1/2))*(x^2+1)^(3/2)*diff(y(x),x) = 1+y(x)^2, y(x),'implicit')

Maple raw output

1/(x^2+1)^(1/2)*x-arcsinh(y(x))-1/2*ln(1+y(x)^2)+_C1 = 0