( x2 + 1) 3∕2 ( y(x) + ∘ ______ y(x)2 + 1) y′(x) = y(x)2 + 1

4.15.31 ${(x^{2} + 1)}^{3 / 2} (y (x) + \sqrt{y (x)^{2} + 1}) y^{'} (x) = y (x)^{2} + 1$

ODE
${(x^{2} + 1)}^{3 / 2} (y (x) + \sqrt{y (x)^{2} + 1}) y^{'} (x) = y (x)^{2} + 1$ ODE Classiﬁcation

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica ✓
cpu = 0.127647 (sec), leaf count = 321

${{y (x) \to - \frac{i (\sinh (c_{1} + \frac{x}{\sqrt{x^{2} + 1}}) + \cosh (c_{1} + \frac{x}{\sqrt{x^{2} + 1}}) - 1)}{\sqrt{- 2 \sinh (c_{1} + \frac{x}{\sqrt{x^{2} + 1}}) - 2 \cosh (c_{1} + \frac{x}{\sqrt{x^{2} + 1}}) + 1}}}, {y (x) \to \frac{i (\sinh (c_{1} + \frac{x}{\sqrt{x^{2} + 1}}) + \cosh (c_{1} + \frac{x}{\sqrt{x^{2} + 1}}) - 1)}{\sqrt{- 2 \sinh (c_{1} + \frac{x}{\sqrt{x^{2} + 1}}) - 2 \cosh (c_{1} + \frac{x}{\sqrt{x^{2} + 1}}) + 1}}}, {y (x) \to - \frac{i (\sinh (c_{1} + \frac{x}{\sqrt{x^{2} + 1}}) + \cosh (c_{1} + \frac{x}{\sqrt{x^{2} + 1}}) + 1)}{\sqrt{2 \sinh (c_{1} + \frac{x}{\sqrt{x^{2} + 1}}) + 2 \cosh (c_{1} + \frac{x}{\sqrt{x^{2} + 1}}) + 1}}}, {y (x) \to \frac{i (\sinh (c_{1} + \frac{x}{\sqrt{x^{2} + 1}}) + \cosh (c_{1} + \frac{x}{\sqrt{x^{2} + 1}}) + 1)}{\sqrt{2 \sinh (c_{1} + \frac{x}{\sqrt{x^{2} + 1}}) + 2 \cosh (c_{1} + \frac{x}{\sqrt{x^{2} + 1}}) + 1}}}}$

Maple ✓
cpu = 0.015 (sec), leaf count = 28

${x \frac{1}{\sqrt{x^{2} + 1}} - A r c s i n h (y (x)) - \frac{\ln (1 + {(y (x))}^{2})}{2} +_C 1 = 0}$ 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

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4.15.31 (x2+1)3/2(y(x)+y(x)2+1)y′(x)=y(x)2+1

4.15.31 ${(x^{2} + 1)}^{3 / 2} (y (x) + \sqrt{y (x)^{2} + 1}) y^{'} (x) = y (x)^{2} + 1$