ODE
\[ y''(x)=a y(x) \left (\left (b-y'(x)\right )^2+1\right )^{3/2} \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 17.9734 (sec), leaf count = 1391
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {i \left (\sqrt {a^2 \left (b^2+1\right )}-a c_1\right ) \sqrt {\frac {-2 a^2 \text {$\#$1}^2-4 a c_1+4 \sqrt {a^2 \left (b^2+1\right )}}{\sqrt {a^2 \left (b^2+1\right )}-a c_1}} \sqrt {\frac {a^2 \text {$\#$1}^2+2 a c_1+2 \sqrt {a^2 \left (b^2+1\right )}}{a c_1+\sqrt {a^2 \left (b^2+1\right )}}} \sqrt {-\left (a \text {$\#$1}^2+2 c_1\right ){}^2 \left (a^2 \text {$\#$1}^4+4 a c_1 \text {$\#$1}^2-4 b^2+4 c_1^2-4\right )} E\left (i \sinh ^{-1}\left (\sqrt {\frac {a^2}{2 a c_1+2 \sqrt {a^2 \left (b^2+1\right )}}} \text {$\#$1}\right )|-\frac {a c_1+\sqrt {a^2 \left (b^2+1\right )}}{\sqrt {a^2 \left (b^2+1\right )}-a c_1}\right )-a b \sqrt {\frac {a^2}{a c_1+\sqrt {a^2 \left (b^2+1\right )}}} \text {$\#$1} \left (a \text {$\#$1}^2+2 c_1\right ) \left (a^2 \text {$\#$1}^4+4 a c_1 \text {$\#$1}^2-4 b^2+4 c_1^2-4\right )-i \sqrt {2} \sqrt {a^2 \left (b^2+1\right )} F\left (i \sinh ^{-1}\left (\sqrt {\frac {a^2}{2 a c_1+2 \sqrt {a^2 \left (b^2+1\right )}}} \text {$\#$1}\right )|-\frac {a c_1+\sqrt {a^2 \left (b^2+1\right )}}{\sqrt {a^2 \left (b^2+1\right )}-a c_1}\right ) \sqrt {\frac {-a^2 \text {$\#$1}^2-2 a c_1+2 \sqrt {a^2 \left (b^2+1\right )}}{\sqrt {a^2 \left (b^2+1\right )}-a c_1}} \sqrt {\frac {a^2 \text {$\#$1}^2+2 a c_1+2 \sqrt {a^2 \left (b^2+1\right )}}{a c_1+\sqrt {a^2 \left (b^2+1\right )}}} \sqrt {-\left (a \text {$\#$1}^2+2 c_1\right ){}^2 \left (a^2 \text {$\#$1}^4+4 a c_1 \text {$\#$1}^2-4 b^2+4 c_1^2-4\right )}}{a \left (b^2+1\right ) \sqrt {\frac {a^2}{a c_1+\sqrt {a^2 \left (b^2+1\right )}}} \left (a \text {$\#$1}^2+2 c_1\right ) \left (a^2 \text {$\#$1}^4+4 a c_1 \text {$\#$1}^2-4 b^2+4 c_1^2-4\right )}\& \right ]\left [x+c_2\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {i \left (\sqrt {a^2 \left (b^2+1\right )}-a c_1\right ) \sqrt {\frac {-2 a^2 \text {$\#$1}^2-4 a c_1+4 \sqrt {a^2 \left (b^2+1\right )}}{\sqrt {a^2 \left (b^2+1\right )}-a c_1}} \sqrt {\frac {a^2 \text {$\#$1}^2+2 a c_1+2 \sqrt {a^2 \left (b^2+1\right )}}{a c_1+\sqrt {a^2 \left (b^2+1\right )}}} \sqrt {-\left (a \text {$\#$1}^2+2 c_1\right ){}^2 \left (a^2 \text {$\#$1}^4+4 a c_1 \text {$\#$1}^2-4 b^2+4 c_1^2-4\right )} E\left (i \sinh ^{-1}\left (\sqrt {\frac {a^2}{2 a c_1+2 \sqrt {a^2 \left (b^2+1\right )}}} \text {$\#$1}\right )|-\frac {a c_1+\sqrt {a^2 \left (b^2+1\right )}}{\sqrt {a^2 \left (b^2+1\right )}-a c_1}\right )+a b \sqrt {\frac {a^2}{a c_1+\sqrt {a^2 \left (b^2+1\right )}}} \text {$\#$1} \left (a \text {$\#$1}^2+2 c_1\right ) \left (a^2 \text {$\#$1}^4+4 a c_1 \text {$\#$1}^2-4 b^2+4 c_1^2-4\right )-i \sqrt {2} \sqrt {a^2 \left (b^2+1\right )} F\left (i \sinh ^{-1}\left (\sqrt {\frac {a^2}{2 a c_1+2 \sqrt {a^2 \left (b^2+1\right )}}} \text {$\#$1}\right )|-\frac {a c_1+\sqrt {a^2 \left (b^2+1\right )}}{\sqrt {a^2 \left (b^2+1\right )}-a c_1}\right ) \sqrt {\frac {-a^2 \text {$\#$1}^2-2 a c_1+2 \sqrt {a^2 \left (b^2+1\right )}}{\sqrt {a^2 \left (b^2+1\right )}-a c_1}} \sqrt {\frac {a^2 \text {$\#$1}^2+2 a c_1+2 \sqrt {a^2 \left (b^2+1\right )}}{a c_1+\sqrt {a^2 \left (b^2+1\right )}}} \sqrt {-\left (a \text {$\#$1}^2+2 c_1\right ){}^2 \left (a^2 \text {$\#$1}^4+4 a c_1 \text {$\#$1}^2-4 b^2+4 c_1^2-4\right )}}{a \left (b^2+1\right ) \sqrt {\frac {a^2}{a c_1+\sqrt {a^2 \left (b^2+1\right )}}} \left (a \text {$\#$1}^2+2 c_1\right ) \left (a^2 \text {$\#$1}^4+4 a c_1 \text {$\#$1}^2-4 b^2+4 c_1^2-4\right )}\& \right ]\left [x+c_2\right ]\right \}\right \}\]
Maple ✓
cpu = 0.247 (sec), leaf count = 198
\[ \left \{ \int ^{y \left ( x \right ) }\!{( \left ( {{\it \_a}}^{2}+2\,{\it \_C1} \right ) ^{2}b{a}^{2}-4\,{b}^{3}) \left ( - \left ( {{\it \_a}}^{2}+2\,{\it \_C1} \right ) a\sqrt {-{b}^{2} \left ( {a}^{2} \left ( {{\it \_a}}^{2}+2\,{\it \_C1} \right ) ^{2}-4\,{b}^{2}-4 \right ) }+{b}^{2} \left ( {a}^{2} \left ( {{\it \_a}}^{2}+2\,{\it \_C1} \right ) ^{2}-4\,{b}^{2}-4 \right ) \right ) ^{-1}}{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!{( \left ( {{\it \_a}}^{2}+2\,{\it \_C1} \right ) ^{2}b{a}^{2}-4\,{b}^{3}) \left ( \left ( {{\it \_a}}^{2}+2\,{\it \_C1} \right ) a\sqrt {-{b}^{2} \left ( {a}^{2} \left ( {{\it \_a}}^{2}+2\,{\it \_C1} \right ) ^{2}-4\,{b}^{2}-4 \right ) }+{b}^{2} \left ( {a}^{2} \left ( {{\it \_a}}^{2}+2\,{\it \_C1} \right ) ^{2}-4\,{b}^{2}-4 \right ) \right ) ^{-1}}{d{\it \_a}}-x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[y''[x] == a*y[x]*(1 + (b - y'[x])^2)^(3/2),y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[-((-(a*b*Sqrt[a^2/(Sqrt[a^2*(1 + b^2)] + a*C[1])]*#1*(2*C[1] + a*#1^2)*(-4 - 4*b^2 + 4*C[1]^2 + 4*a*C[1]*#1^2 + a^2*#1^4)) + I*(Sqrt[a^2*(1 + b^2)] - a*C[1])*EllipticE[I*ArcSinh[Sqrt[a^2/(2*Sqrt[a^2*(1 + b^2)] + 2*a*C[1])]*#1], -((Sqrt[a^2*(1 + b^2)] + a*C[1])/(Sqrt[a^2*(1 + b^2)] - a*C[1]))]*Sqrt[(4*Sqrt[a^2*(1 + b^2)] - 4*a*C[1] - 2*a^2*#1^2)/(Sqrt[a^2*(1 + b^2)] - a*C[1])]*Sqrt[(2*Sqrt[a^2*(1 + b^2)] + 2*a*C[1] + a^2*#1^2)/(Sqrt[a^2*(1 + b^2)] + a*C[1])]*Sqrt[-((2*C[1] + a*#1^2)^2*(-4 - 4*b^2 + 4*C[1]^2 + 4*a*C[1]*#1^2 + a^2*#1^4))] - I*Sqrt[2]*Sqrt[a^2*(1 + b^2)]*EllipticF[I*ArcSinh[Sqrt[a^2/(2*Sqrt[a^2*(1 + b^2)] + 2*a*C[1])]*#1], -((Sqrt[a^2*(1 + b^2)] + a*C[1])/(Sqrt[a^2*(1 + b^2)] - a*C[1]))]*Sqrt[(2*Sqrt[a^2*(1 + b^2)] - 2*a*C[1] - a^2*#1^2)/(Sqrt[a^2*(1 + b^2)] - a*C[1])]*Sqrt[(2*Sqrt[a^2*(1 + b^2)] + 2*a*C[1] + a^2*#1^2)/(Sqrt[a^2*(1 + b^2)] + a*C[1])]*Sqrt[-((2*C[1] + a*#1^2)^2*(-4 - 4*b^2 + 4*C[1]^2 + 4*a*C[1]*#1^2 + a^2*#1^4))])/(a*(1 + b^2)*Sqrt[a^2/(Sqrt[a^2*(1 + b^2)] + a*C[1])]*(2*C[1] + a*#1^2)*(-4 - 4*b^2 + 4*C[1]^2 + 4*a*C[1]*#1^2 + a^2*#1^4))) & ][x + C[2]]}, {y[x] -> InverseFunction[(a*b*Sqrt[a^2/(Sqrt[a^2*(1 + b^2)] + a*C[1])]*#1*(2*C[1] + a*#1^2)*(-4 - 4*b^2 + 4*C[1]^2 + 4*a*C[1]*#1^2 + a^2*#1^4) + I*(Sqrt[a^2*(1 + b^2)] - a*C[1])*EllipticE[I*ArcSinh[Sqrt[a^2/(2*Sqrt[a^2*(1 + b^2)] + 2*a*C[1])]*#1], -((Sqrt[a^2*(1 + b^2)] + a*C[1])/(Sqrt[a^2*(1 + b^2)] - a*C[1]))]*Sqrt[(4*Sqrt[a^2*(1 + b^2)] - 4*a*C[1] - 2*a^2*#1^2)/(Sqrt[a^2*(1 + b^2)] - a*C[1])]*Sqrt[(2*Sqrt[a^2*(1 + b^2)] + 2*a*C[1] + a^2*#1^2)/(Sqrt[a^2*(1 + b^2)] + a*C[1])]*Sqrt[-((2*C[1] + a*#1^2)^2*(-4 - 4*b^2 + 4*C[1]^2 + 4*a*C[1]*#1^2 + a^2*#1^4))] - I*Sqrt[2]*Sqrt[a^2*(1 + b^2)]*EllipticF[I*ArcSinh[Sqrt[a^2/(2*Sqrt[a^2*(1 + b^2)] + 2*a*C[1])]*#1], -((Sqrt[a^2*(1 + b^2)] + a*C[1])/(Sqrt[a^2*(1 + b^2)] - a*C[1]))]*Sqrt[(2*Sqrt[a^2*(1 + b^2)] - 2*a*C[1] - a^2*#1^2)/(Sqrt[a^2*(1 + b^2)] - a*C[1])]*Sqrt[(2*Sqrt[a^2*(1 + b^2)] + 2*a*C[1] + a^2*#1^2)/(Sqrt[a^2*(1 + b^2)] + a*C[1])]*Sqrt[-((2*C[1] + a*#1^2)^2*(-4 - 4*b^2 + 4*C[1]^2 + 4*a*C[1]*#1^2 + a^2*#1^4))])/(a*(1 + b^2)*Sqrt[a^2/(Sqrt[a^2*(1 + b^2)] + a*C[1])]*(2*C[1] + a*#1^2)*(-4 - 4*b^2 + 4*C[1]^2 + 4*a*C[1]*#1^2 + a^2*#1^4)) & ][x + C[2]]}}
Maple raw input
dsolve(diff(diff(y(x),x),x) = a*y(x)*(1+(b-diff(y(x),x))^2)^(3/2), y(x),'implicit')
Maple raw output
Intat(((_a^2+2*_C1)^2*b*a^2-4*b^3)/((_a^2+2*_C1)*a*(-b^2*(a^2*(_a^2+2*_C1)^2-4*b^2-4))^(1/2)+b^2*(a^2*(_a^2+2*_C1)^2-4*b^2-4)),_a = y(x))-x-_C2 = 0, Intat(((_a^2+2*_C1)^2*b*a^2-4*b^3)/(-(_a^2+2*_C1)*a*(-b^2*(a^2*(_a^2+2*_C1)^2-4*b^2-4))^(1/2)+b^2*(a^2*(_a^2+2*_C1)^2-4*b^2-4)),_a = y(x))-x-_C2 = 0