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 = 0.83103 (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 = 1.937 (sec), leaf count = 337
\[\left [y \left (x \right ) = \int \left (\frac {\sqrt {\textit {\_C1}^{2} a^{2} b^{4}+2 \textit {\_C1} \,a^{2} b^{4} x +a^{2} b^{4} x^{2}-1}\, x}{2 b}+\frac {\sqrt {\textit {\_C1}^{2} a^{2} b^{4}+2 \textit {\_C1} \,a^{2} b^{4} x +a^{2} b^{4} x^{2}-1}\, \textit {\_C1}}{2 b}-\frac {\ln \left (\frac {\textit {\_C1} \,a^{2} b^{4}+a^{2} b^{4} x}{\sqrt {a^{2} b^{4}}}+\sqrt {\textit {\_C1}^{2} a^{2} b^{4}+2 \textit {\_C1} \,a^{2} b^{4} x +a^{2} b^{4} x^{2}-1}\right )}{2 b \sqrt {a^{2} b^{4}}}\right )d x +\textit {\_C2} x +\textit {\_C3}, y \left (x \right ) = \int \left (-\frac {\sqrt {\textit {\_C1}^{2} a^{2} b^{4}+2 \textit {\_C1} \,a^{2} b^{4} x +a^{2} b^{4} x^{2}-1}\, x}{2 b}-\frac {\sqrt {\textit {\_C1}^{2} a^{2} b^{4}+2 \textit {\_C1} \,a^{2} b^{4} x +a^{2} b^{4} x^{2}-1}\, \textit {\_C1}}{2 b}+\frac {\ln \left (\frac {\textit {\_C1} \,a^{2} b^{4}+a^{2} b^{4} x}{\sqrt {a^{2} b^{4}}}+\sqrt {\textit {\_C1}^{2} a^{2} b^{4}+2 \textit {\_C1} \,a^{2} b^{4} x +a^{2} b^{4} x^{2}-1}\right )}{2 b \sqrt {a^{2} b^{4}}}\right )d x +\textit {\_C2} x +\textit {\_C3}\right ]\] 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))
Maple raw output
[y(x) = Int(1/2/b*(_C1^2*a^2*b^4+2*_C1*a^2*b^4*x+a^2*b^4*x^2-1)^(1/2)*x+1/2/b*(_C1^2*a^2*b^4+2*_C1*a^2*b^4*x+a^2*b^4*x^2-1)^(1/2)*_C1-1/2/b*ln((_C1*a^2*b^4+a^2*b^4*x)/(a^2*b^4)^(1/2)+(_C1^2*a^2*b^4+2*_C1*a^2*b^4*x+a^2*b^4*x^2-1)^(1/2))/(a^2*b^4)^(1/2),x)+_C2*x+_C3, y(x) = Int(-1/2/b*(_C1^2*a^2*b^4+2*_C1*a^2*b^4*x+a^2*b^4*x^2-1)^(1/2)*x-1/2/b*(_C1^2*a^2*b^4+2*_C1*a^2*b^4*x+a^2*b^4*x^2-1)^(1/2)*_C1+1/2/b*ln((_C1*a^2*b^4+a^2*b^4*x)/(a^2*b^4)^(1/2)+(_C1^2*a^2*b^4+2*_C1*a^2*b^4*x+a^2*b^4*x^2-1)^(1/2))/(a^2*b^4)^(1/2),x)+_C2*x+_C3]