ODE
\[ y'(x)^2=a^2-y(x)^2 \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Independent variable missing, Solve for \(y'\)
Mathematica ✓
cpu = 0.0446303 (sec), leaf count = 99
\[\left \{\left \{y(x)\to -\frac {a \tan \left (x-c_1\right )}{\sqrt {\sec ^2\left (x-c_1\right )}}\right \},\left \{y(x)\to \frac {a \tan \left (x-c_1\right )}{\sqrt {\sec ^2\left (x-c_1\right )}}\right \},\left \{y(x)\to -\frac {a \tan \left (c_1+x\right )}{\sqrt {\sec ^2\left (c_1+x\right )}}\right \},\left \{y(x)\to \frac {a \tan \left (c_1+x\right )}{\sqrt {\sec ^2\left (c_1+x\right )}}\right \}\right \}\]
Maple ✓
cpu = 0.047 (sec), leaf count = 61
\[ \left \{ \left ( y \left ( x \right ) \right ) ^{2}-{a}^{2}=0,x-\arctan \left ( {y \left ( x \right ) {\frac {1}{\sqrt {{a}^{2}- \left ( y \left ( x \right ) \right ) ^{2}}}}} \right ) -{\it \_C1}=0,x+\arctan \left ( {y \left ( x \right ) {\frac {1}{\sqrt {{a}^{2}- \left ( y \left ( x \right ) \right ) ^{2}}}}} \right ) -{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[y'[x]^2 == a^2 - y[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> -((a*Tan[x - C[1]])/Sqrt[Sec[x - C[1]]^2])}, {y[x] -> (a*Tan[x - C[1]])/Sqrt[Sec[x - C[1]]^2]}, {y[x] -> -((a*Tan[x + C[1]])/Sqrt[Sec[x + C[1]]^2])}, {y[x] -> (a*Tan[x + C[1]])/Sqrt[Sec[x + C[1]]^2]}}
Maple raw input
dsolve(diff(y(x),x)^2 = a^2-y(x)^2, y(x),'implicit')
Maple raw output
y(x)^2-a^2 = 0, x-arctan(y(x)/(a^2-y(x)^2)^(1/2))-_C1 = 0, x+arctan(y(x)/(a^2-y(x)^2)^(1/2))-_C1 = 0