ODE
\[ y''(x) y'''(x)=a \sqrt {b^2 y''(x)^2+1} \] ODE Classification
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.531415 (sec), leaf count = 415
\[\left \{\left \{y(x)\to \frac {6 a^2 b^5 c_3 x+6 a^2 b^5 c_2+\left (a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1^2-1\right ){}^{3/2}+3 \sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1^2-1}-3 b^2 c_1 \log \left (\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1^2-1}+a b^2 x+b^2 c_1\right )-3 a b^2 x \log \left (b^2 \left (\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1^2-1}+a b^2 x+b^2 c_1\right )\right )}{6 a^2 b^5}\right \},\left \{y(x)\to \frac {-\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1^2-1} \left (a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1^2+2\right )+3 b^2 c_1 \log \left (\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1^2-1}+a b^2 x+b^2 c_1\right )+3 a b^2 x \log \left (b^2 \left (\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1^2-1}+a b^2 x+b^2 c_1\right )\right )}{6 a^2 b^5}+c_3 x+c_2\right \}\right \}\]
Maple ✓
cpu = 0.192 (sec), leaf count = 197
\[ \left \{ y \left ( x \right ) ={\it \_C2}\,x+\int \!{\frac {1}{2\,b} \left ( -{1\ln \left ( \sqrt { \left ( -1+{b}^{2} \left ( {\it \_C1}+x \right ) a \right ) \left ( 1+{b}^{2} \left ( {\it \_C1}+x \right ) a \right ) }+{ \left ( {\it \_C1}+x \right ) {b}^{4}{a}^{2}{\frac {1}{\sqrt {{a}^{2}{b}^{4}}}}} \right ) {\frac {1}{\sqrt {{a}^{2}{b}^{4}}}}}+\sqrt { \left ( -1+{b}^{2} \left ( {\it \_C1}+x \right ) a \right ) \left ( 1+{b}^{2} \left ( {\it \_C1}+x \right ) a \right ) } \left ( {\it \_C1}+x \right ) \right ) }\,{\rm d}x+{\it \_C3},y \left ( x \right ) ={\it \_C2}\,x+\int \!{\frac {1}{2\,b} \left ( {1\ln \left ( \sqrt { \left ( -1+{b}^{2} \left ( {\it \_C1}+x \right ) a \right ) \left ( 1+{b}^{2} \left ( {\it \_C1}+x \right ) a \right ) }+{ \left ( {\it \_C1}+x \right ) {b}^{4}{a}^{2}{\frac {1}{\sqrt {{a}^{2}{b}^{4}}}}} \right ) {\frac {1}{\sqrt {{a}^{2}{b}^{4}}}}}-\sqrt { \left ( -1+{b}^{2} \left ( {\it \_C1}+x \right ) a \right ) \left ( 1+{b}^{2} \left ( {\it \_C1}+x \right ) a \right ) } \left ( {\it \_C1}+x \right ) \right ) }\,{\rm d}x+{\it \_C3} \right \} \] Mathematica raw input
DSolve[y''[x]*y'''[x] == a*Sqrt[1 + b^2*y''[x]^2],y[x],x]
Mathematica raw output
{{y[x] -> (3*Sqrt[-1 + a^2*b^4*x^2 + 2*a*b^4*x*C[1] + b^4*C[1]^2] + (-1 + a^2*b^4*x^2 + 2*a*b^4*x*C[1] + b^4*C[1]^2)^(3/2) + 6*a^2*b^5*C[2] + 6*a^2*b^5*x*C[3] - 3*b^2*C[1]*Log[a*b^2*x + b^2*C[1] + Sqrt[-1 + a^2*b^4*x^2 + 2*a*b^4*x*C[1] + b^4*C[1]^2]] - 3*a*b^2*x*Log[b^2*(a*b^2*x + b^2*C[1] + Sqrt[-1 + a^2*b^4*x^2 + 2*a*b^4*x*C[1] + b^4*C[1]^2])])/(6*a^2*b^5)}, {y[x] -> C[2] + x*C[3] + (-(Sqrt[-1 + a^2*b^4*x^2 + 2*a*b^4*x*C[1] + b^4*C[1]^2]*(2 + a^2*b^4*x^2 + 2*a*b^4*x*C[1] + b^4*C[1]^2)) + 3*b^2*C[1]*Log[a*b^2*x + b^2*C[1] + Sqrt[-1 + a^2*b^4*x^2 + 2*a*b^4*x*C[1] + b^4*C[1]^2]] + 3*a*b^2*x*Log[b^2*(a*b^2*x + b^2*C[1] + Sqrt[-1 + a^2*b^4*x^2 + 2*a*b^4*x*C[1] + b^4*C[1]^2])])/(6*a^2*b^5)}}
Maple raw input
dsolve(diff(diff(y(x),x),x)*diff(diff(diff(y(x),x),x),x) = a*(1+b^2*diff(diff(y(x),x),x)^2)^(1/2), y(x),'implicit')
Maple raw output
y(x) = _C2*x+Int(1/2*(-ln(((-1+b^2*(_C1+x)*a)*(1+b^2*(_C1+x)*a))^(1/2)+(_C1+x)*b^4*a^2/(a^2*b^4)^(1/2))/(a^2*b^4)^(1/2)+((-1+b^2*(_C1+x)*a)*(1+b^2*(_C1+x)*a))^(1/2)*(_C1+x))/b,x)+_C3, y(x) = _C2*x+Int(1/2*(ln(((-1+b^2*(_C1+x)*a)*(1+b^2*(_C1+x)*a))^(1/2)+(_C1+x)*b^4*a^2/(a^2*b^4)^(1/2))/(a^2*b^4)^(1/2)-((-1+b^2*(_C1+x)*a)*(1+b^2*(_C1+x)*a))^(1/2)*(_C1+x))/b,x)+_C3