4.15.25 \(\sqrt {b^2+y(x)^2} y'(x)=\sqrt {a^2+x^2}\)

ODE
\[ \sqrt {b^2+y(x)^2} y'(x)=\sqrt {a^2+x^2} \] ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.17614 (sec), leaf count = 89

\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {1}{2} \text {$\#$1} \sqrt {\text {$\#$1}^2+b^2}+\frac {1}{2} b^2 \log \left (\sqrt {\text {$\#$1}^2+b^2}+\text {$\#$1}\right )\& \right ]\left [\frac {1}{2} x \sqrt {a^2+x^2}+\frac {1}{2} a^2 \log \left (\sqrt {a^2+x^2}+x\right )+c_1\right ]\right \}\right \}\]

Maple
cpu = 0.013 (sec), leaf count = 67

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

DSolve[Sqrt[b^2 + y[x]^2]*y'[x] == Sqrt[a^2 + x^2],y[x],x]

Mathematica raw output

{{y[x] -> InverseFunction[(b^2*Log[#1 + Sqrt[b^2 + #1^2]])/2 + (#1*Sqrt[b^2 + #1
^2])/2 & ][(x*Sqrt[a^2 + x^2])/2 + C[1] + (a^2*Log[x + Sqrt[a^2 + x^2]])/2]}}

Maple raw input

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

Maple raw output

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