ODE
\[ \left (a^2-y(x)^2\right ) y'(x)^2=y(x)^2 \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Independent variable missing, Solve for \(y'\)
Mathematica ✓
cpu = 0.264288 (sec), leaf count = 111
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\sqrt {a^2-\text {$\#$1}^2}-a \log \left (a \sqrt {a^2-\text {$\#$1}^2}+a^2\right )+a \log (\text {$\#$1})\& \right ]\left [c_1-x\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\sqrt {a^2-\text {$\#$1}^2}-a \log \left (a \sqrt {a^2-\text {$\#$1}^2}+a^2\right )+a \log (\text {$\#$1})\& \right ]\left [c_1+x\right ]\right \}\right \}\]
Maple ✓
cpu = 0.497 (sec), leaf count = 122
\[ \left \{ x-\sqrt {{a}^{2}- \left ( y \left ( x \right ) \right ) ^{2}}+{{a}^{2}\ln \left ( {\frac {1}{y \left ( x \right ) } \left ( 2\,{a}^{2}+2\,\sqrt {{a}^{2}}\sqrt {{a}^{2}- \left ( y \left ( x \right ) \right ) ^{2}} \right ) } \right ) {\frac {1}{\sqrt {{a}^{2}}}}}-{\it \_C1}=0,x+\sqrt {{a}^{2}- \left ( y \left ( x \right ) \right ) ^{2}}-{{a}^{2}\ln \left ( {\frac {1}{y \left ( x \right ) } \left ( 2\,{a}^{2}+2\,\sqrt {{a}^{2}}\sqrt {{a}^{2}- \left ( y \left ( x \right ) \right ) ^{2}} \right ) } \right ) {\frac {1}{\sqrt {{a}^{2}}}}}-{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[(a^2 - y[x]^2)*y'[x]^2 == y[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[a*Log[#1] - a*Log[a^2 + a*Sqrt[a^2 - #1^2]] + Sqrt[a^2 - #1^2] & ][-x + C[1]]}, {y[x] -> InverseFunction[a*Log[#1] - a*Log[a^2 + a*Sqrt[a^2 - #1^2]] + Sqrt[a^2 - #1^2] & ][x + C[1]]}}
Maple raw input
dsolve((a^2-y(x)^2)*diff(y(x),x)^2 = y(x)^2, y(x),'implicit')
Maple raw output
x-(a^2-y(x)^2)^(1/2)+a^2/(a^2)^(1/2)*ln((2*a^2+2*(a^2)^(1/2)*(a^2-y(x)^2)^(1/2))/y(x))-_C1 = 0, x+(a^2-y(x)^2)^(1/2)-a^2/(a^2)^(1/2)*ln((2*a^2+2*(a^2)^(1/2)*(a^2-y(x)^2)^(1/2))/y(x))-_C1 = 0