ODE
\[ y''(x)=a \left (y'(x)^2+1\right )^{3/2} (b+c x+y(x)) \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 38.3644 (sec), leaf count = 1760
\[\left \{\text {Solve}\left [c_2=\int _1^x\frac {a^2 y(x)^4+4 a^2 (b+c K[1]) y(x)^3+2 a^2 \left (3 b^2+6 c K[1] b+3 c^2 K[1]^2-2 c_1\right ) y(x)^2+4 a^2 (b+c K[1]) \left (b^2+2 c K[1] b+c^2 K[1]^2-2 c_1\right ) y(x)-4 c^2+a^2 \left (b^2+2 c K[1] b+c^2 K[1]^2-2 c_1\right ){}^2-c \sqrt {-a^2 \left (b^2+2 c K[1] b+c^2 K[1]^2+y(x)^2-2 c_1+2 (b+c K[1]) y(x)\right ){}^2 \left (a^2 y(x)^4+4 a^2 (b+c K[1]) y(x)^3+2 a^2 \left (3 b^2+6 c K[1] b+3 c^2 K[1]^2-2 c_1\right ) y(x)^2+4 a^2 (b+c K[1]) \left (b^2+2 c K[1] b+c^2 K[1]^2-2 c_1\right ) y(x)+a^2 \left (b^2+2 c K[1] b+c^2 K[1]^2-2 c_1\right ){}^2-4 \left (c^2+1\right )\right )}-4}{\left (c^2+1\right ) \left (a^2 y(x)^4+4 a^2 (b+c K[1]) y(x)^3+2 a^2 \left (3 b^2+6 c K[1] b+3 c^2 K[1]^2-2 c_1\right ) y(x)^2+4 a^2 (b+c K[1]) \left (b^2+2 c K[1] b+c^2 K[1]^2-2 c_1\right ) y(x)+a^2 \left (b^2+2 c K[1] b+c^2 K[1]^2-2 c_1\right ){}^2-4 \left (c^2+1\right )\right )}dK[1]+\int _1^{y(x)}\left (-\frac {c}{c^2+1}-\int _1^x-\frac {8 a^2 c (b+c K[1]+K[2]) \left (b^2+2 (c K[1]+K[2]) b+(c K[1]+K[2])^2-2 c_1\right )}{\left (a^2 \left (b^2+2 (c K[1]+K[2]) b+(c K[1]+K[2])^2-2 c_1\right ){}^2-4 \left (c^2+1\right )\right ) \sqrt {4 a^2 \left (c^2+1\right ) \left (b^2+2 (c K[1]+K[2]) b+(c K[1]+K[2])^2-2 c_1\right ){}^2-a^4 \left (b^2+2 (c K[1]+K[2]) b+(c K[1]+K[2])^2-2 c_1\right ){}^4}}dK[1]-\frac {\sqrt {4 a^2 \left (c^2+1\right ) \left (b^2+2 (c x+K[2]) b+c^2 x^2+K[2]^2-2 c_1+2 c x K[2]\right ){}^2-a^4 \left (b^2+2 (c x+K[2]) b+c^2 x^2+K[2]^2-2 c_1+2 c x K[2]\right ){}^4}}{\left (c^2+1\right ) \left (a^2 \left (b^2+2 (c x+K[2]) b+c^2 x^2+K[2]^2-2 c_1+2 c x K[2]\right ){}^2-4 \left (c^2+1\right )\right )}\right )dK[2],y(x)\right ],\text {Solve}\left [c_2=\int _1^x\frac {a^2 y(x)^4+4 a^2 (b+c K[3]) y(x)^3+2 a^2 \left (3 b^2+6 c K[3] b+3 c^2 K[3]^2-2 c_1\right ) y(x)^2+4 a^2 (b+c K[3]) \left (b^2+2 c K[3] b+c^2 K[3]^2-2 c_1\right ) y(x)-4 c^2+a^2 \left (b^2+2 c K[3] b+c^2 K[3]^2-2 c_1\right ){}^2+c \sqrt {-a^2 \left (b^2+2 c K[3] b+c^2 K[3]^2+y(x)^2-2 c_1+2 (b+c K[3]) y(x)\right ){}^2 \left (a^2 y(x)^4+4 a^2 (b+c K[3]) y(x)^3+2 a^2 \left (3 b^2+6 c K[3] b+3 c^2 K[3]^2-2 c_1\right ) y(x)^2+4 a^2 (b+c K[3]) \left (b^2+2 c K[3] b+c^2 K[3]^2-2 c_1\right ) y(x)+a^2 \left (b^2+2 c K[3] b+c^2 K[3]^2-2 c_1\right ){}^2-4 \left (c^2+1\right )\right )}-4}{\left (c^2+1\right ) \left (a^2 y(x)^4+4 a^2 (b+c K[3]) y(x)^3+2 a^2 \left (3 b^2+6 c K[3] b+3 c^2 K[3]^2-2 c_1\right ) y(x)^2+4 a^2 (b+c K[3]) \left (b^2+2 c K[3] b+c^2 K[3]^2-2 c_1\right ) y(x)+a^2 \left (b^2+2 c K[3] b+c^2 K[3]^2-2 c_1\right ){}^2-4 \left (c^2+1\right )\right )}dK[3]+\int _1^{y(x)}-\frac {\int _1^x\frac {8 a^2 c (b+c K[3]+K[4]) \left (b^2+2 (c K[3]+K[4]) b+(c K[3]+K[4])^2-2 c_1\right )}{\left (a^2 \left (b^2+2 (c K[3]+K[4]) b+(c K[3]+K[4])^2-2 c_1\right ){}^2-4 \left (c^2+1\right )\right ) \sqrt {4 a^2 \left (c^2+1\right ) \left (b^2+2 (c K[3]+K[4]) b+(c K[3]+K[4])^2-2 c_1\right ){}^2-a^4 \left (b^2+2 (c K[3]+K[4]) b+(c K[3]+K[4])^2-2 c_1\right ){}^4}}dK[3] c^2+c+\int _1^x\frac {8 a^2 c (b+c K[3]+K[4]) \left (b^2+2 (c K[3]+K[4]) b+(c K[3]+K[4])^2-2 c_1\right )}{\left (a^2 \left (b^2+2 (c K[3]+K[4]) b+(c K[3]+K[4])^2-2 c_1\right ){}^2-4 \left (c^2+1\right )\right ) \sqrt {4 a^2 \left (c^2+1\right ) \left (b^2+2 (c K[3]+K[4]) b+(c K[3]+K[4])^2-2 c_1\right ){}^2-a^4 \left (b^2+2 (c K[3]+K[4]) b+(c K[3]+K[4])^2-2 c_1\right ){}^4}}dK[3]-\frac {\sqrt {4 a^2 \left (c^2+1\right ) \left (b^2+2 (c x+K[4]) b+c^2 x^2+K[4]^2-2 c_1+2 c x K[4]\right ){}^2-a^4 \left (b^2+2 (c x+K[4]) b+c^2 x^2+K[4]^2-2 c_1+2 c x K[4]\right ){}^4}}{a^2 \left (b^2+2 (c x+K[4]) b+c^2 x^2+K[4]^2-2 c_1+2 c x K[4]\right ){}^2-4 \left (c^2+1\right )}}{c^2+1}dK[4],y(x)\right ]\right \}\]
Maple ✓
cpu = 1.563 (sec), leaf count = 771
\[\left [y \left (x \right ) = -c x +\RootOf \left (-x +\int _{}^{\textit {\_Z}}\frac {4 a^{2} b^{2} c^{2} \textit {\_f}^{2}+4 a^{2} b \,c^{2} \textit {\_f}^{3}+a^{2} c^{2} \textit {\_f}^{4}-8 \textit {\_C1} \,a^{2} b \,c^{2} \textit {\_f} -4 \textit {\_C1} \,a^{2} c^{2} \textit {\_f}^{2}+4 \textit {\_C1}^{2} a^{2} c^{2}-2 \sqrt {-c^{2} \left (\textit {\_f}^{4} a^{2}+4 \textit {\_f}^{3} a^{2} b +4 \textit {\_f}^{2} a^{2} b^{2}-4 \textit {\_C1} \,\textit {\_f}^{2} a^{2}-8 \textit {\_C1} \textit {\_f} \,a^{2} b +4 a^{2} \textit {\_C1}^{2}-4 c^{2}-4\right )}\, a b \textit {\_f} -\sqrt {-c^{2} \left (\textit {\_f}^{4} a^{2}+4 \textit {\_f}^{3} a^{2} b +4 \textit {\_f}^{2} a^{2} b^{2}-4 \textit {\_C1} \,\textit {\_f}^{2} a^{2}-8 \textit {\_C1} \textit {\_f} \,a^{2} b +4 a^{2} \textit {\_C1}^{2}-4 c^{2}-4\right )}\, a \,\textit {\_f}^{2}-4 c^{4}+2 \sqrt {-c^{2} \left (\textit {\_f}^{4} a^{2}+4 \textit {\_f}^{3} a^{2} b +4 \textit {\_f}^{2} a^{2} b^{2}-4 \textit {\_C1} \,\textit {\_f}^{2} a^{2}-8 \textit {\_C1} \textit {\_f} \,a^{2} b +4 a^{2} \textit {\_C1}^{2}-4 c^{2}-4\right )}\, \textit {\_C1} a -4 c^{2}}{\left (\textit {\_f}^{4} a^{2}+4 \textit {\_f}^{3} a^{2} b +4 \textit {\_f}^{2} a^{2} b^{2}-4 \textit {\_C1} \,\textit {\_f}^{2} a^{2}-8 \textit {\_C1} \textit {\_f} \,a^{2} b +4 a^{2} \textit {\_C1}^{2}-4 c^{2}-4\right ) \left (c^{2}+1\right ) c}d \textit {\_f} +\textit {\_C2} \right ), y \left (x \right ) = -c x +\RootOf \left (-x +\int _{}^{\textit {\_Z}}-\frac {-a^{2} c^{2} \textit {\_f}^{4}-4 a^{2} b \,c^{2} \textit {\_f}^{3}-4 a^{2} b^{2} c^{2} \textit {\_f}^{2}+4 \textit {\_C1} \,a^{2} c^{2} \textit {\_f}^{2}+8 \textit {\_C1} \,a^{2} b \,c^{2} \textit {\_f} -4 \textit {\_C1}^{2} a^{2} c^{2}-\sqrt {-c^{2} \left (\textit {\_f}^{4} a^{2}+4 \textit {\_f}^{3} a^{2} b +4 \textit {\_f}^{2} a^{2} b^{2}-4 \textit {\_C1} \,\textit {\_f}^{2} a^{2}-8 \textit {\_C1} \textit {\_f} \,a^{2} b +4 a^{2} \textit {\_C1}^{2}-4 c^{2}-4\right )}\, a \,\textit {\_f}^{2}-2 \sqrt {-c^{2} \left (\textit {\_f}^{4} a^{2}+4 \textit {\_f}^{3} a^{2} b +4 \textit {\_f}^{2} a^{2} b^{2}-4 \textit {\_C1} \,\textit {\_f}^{2} a^{2}-8 \textit {\_C1} \textit {\_f} \,a^{2} b +4 a^{2} \textit {\_C1}^{2}-4 c^{2}-4\right )}\, a b \textit {\_f} +4 c^{4}+2 \sqrt {-c^{2} \left (\textit {\_f}^{4} a^{2}+4 \textit {\_f}^{3} a^{2} b +4 \textit {\_f}^{2} a^{2} b^{2}-4 \textit {\_C1} \,\textit {\_f}^{2} a^{2}-8 \textit {\_C1} \textit {\_f} \,a^{2} b +4 a^{2} \textit {\_C1}^{2}-4 c^{2}-4\right )}\, \textit {\_C1} a +4 c^{2}}{\left (\textit {\_f}^{4} a^{2}+4 \textit {\_f}^{3} a^{2} b +4 \textit {\_f}^{2} a^{2} b^{2}-4 \textit {\_C1} \,\textit {\_f}^{2} a^{2}-8 \textit {\_C1} \textit {\_f} \,a^{2} b +4 a^{2} \textit {\_C1}^{2}-4 c^{2}-4\right ) \left (c^{2}+1\right ) c}d \textit {\_f} +\textit {\_C2} \right )\right ]\] Mathematica raw input
DSolve[y''[x] == a*(b + c*x + y[x])*(1 + y'[x]^2)^(3/2),y[x],x]
Mathematica raw output
{Solve[C[2] == Inactive[Integrate][(-4 - 4*c^2 + a^2*(b^2 - 2*C[1] + 2*b*c*K[1] + c^2*K[1]^2)^2 + 4*a^2*(b + c*K[1])*(b^2 - 2*C[1] + 2*b*c*K[1] + c^2*K[1]^2)*y[x] + 2*a^2*(3*b^2 - 2*C[1] + 6*b*c*K[1] + 3*c^2*K[1]^2)*y[x]^2 + 4*a^2*(b + c*K[1])*y[x]^3 + a^2*y[x]^4 - c*Sqrt[-(a^2*(b^2 - 2*C[1] + 2*b*c*K[1] + c^2*K[1]^2 + 2*(b + c*K[1])*y[x] + y[x]^2)^2*(-4*(1 + c^2) + a^2*(b^2 - 2*C[1] + 2*b*c*K[1] + c^2*K[1]^2)^2 + 4*a^2*(b + c*K[1])*(b^2 - 2*C[1] + 2*b*c*K[1] + c^2*K[1]^2)*y[x] + 2*a^2*(3*b^2 - 2*C[1] + 6*b*c*K[1] + 3*c^2*K[1]^2)*y[x]^2 + 4*a^2*(b + c*K[1])*y[x]^3 + a^2*y[x]^4))])/((1 + c^2)*(-4*(1 + c^2) + a^2*(b^2 - 2*C[1] + 2*b*c*K[1] + c^2*K[1]^2)^2 + 4*a^2*(b + c*K[1])*(b^2 - 2*C[1] + 2*b*c*K[1] + c^2*K[1]^2)*y[x] + 2*a^2*(3*b^2 - 2*C[1] + 6*b*c*K[1] + 3*c^2*K[1]^2)*y[x]^2 + 4*a^2*(b + c*K[1])*y[x]^3 + a^2*y[x]^4)), {K[1], 1, x}] + Inactive[Integrate][-(c/(1 + c^2)) - Sqrt[4*a^2*(1 + c^2)*(b^2 + c^2*x^2 - 2*C[1] + 2*c*x*K[2] + K[2]^2 + 2*b*(c*x + K[2]))^2 - a^4*(b^2 + c^2*x^2 - 2*C[1] + 2*c*x*K[2] + K[2]^2 + 2*b*(c*x + K[2]))^4]/((1 + c^2)*(-4*(1 + c^2) + a^2*(b^2 + c^2*x^2 - 2*C[1] + 2*c*x*K[2] + K[2]^2 + 2*b*(c*x + K[2]))^2)) - Inactive[Integrate][(-8*a^2*c*(b + c*K[1] + K[2])*(b^2 - 2*C[1] + 2*b*(c*K[1] + K[2]) + (c*K[1] + K[2])^2))/((-4*(1 + c^2) + a^2*(b^2 - 2*C[1] + 2*b*(c*K[1] + K[2]) + (c*K[1] + K[2])^2)^2)*Sqrt[4*a^2*(1 + c^2)*(b^2 - 2*C[1] + 2*b*(c*K[1] + K[2]) + (c*K[1] + K[2])^2)^2 - a^4*(b^2 - 2*C[1] + 2*b*(c*K[1] + K[2]) + (c*K[1] + K[2])^2)^4]), {K[1], 1, x}], {K[2], 1, y[x]}], y[x]], Solve[C[2] == Inactive[Integrate][(-4 - 4*c^2 + a^2*(b^2 - 2*C[1] + 2*b*c*K[3] + c^2*K[3]^2)^2 + 4*a^2*(b + c*K[3])*(b^2 - 2*C[1] + 2*b*c*K[3] + c^2*K[3]^2)*y[x] + 2*a^2*(3*b^2 - 2*C[1] + 6*b*c*K[3] + 3*c^2*K[3]^2)*y[x]^2 + 4*a^2*(b + c*K[3])*y[x]^3 + a^2*y[x]^4 + c*Sqrt[-(a^2*(b^2 - 2*C[1] + 2*b*c*K[3] + c^2*K[3]^2 + 2*(b + c*K[3])*y[x] + y[x]^2)^2*(-4*(1 + c^2) + a^2*(b^2 - 2*C[1] + 2*b*c*K[3] + c^2*K[3]^2)^2 + 4*a^2*(b + c*K[3])*(b^2 - 2*C[1] + 2*b*c*K[3] + c^2*K[3]^2)*y[x] + 2*a^2*(3*b^2 - 2*C[1] + 6*b*c*K[3] + 3*c^2*K[3]^2)*y[x]^2 + 4*a^2*(b + c*K[3])*y[x]^3 + a^2*y[x]^4))])/((1 + c^2)*(-4*(1 + c^2) + a^2*(b^2 - 2*C[1] + 2*b*c*K[3] + c^2*K[3]^2)^2 + 4*a^2*(b + c*K[3])*(b^2 - 2*C[1] + 2*b*c*K[3] + c^2*K[3]^2)*y[x] + 2*a^2*(3*b^2 - 2*C[1] + 6*b*c*K[3] + 3*c^2*K[3]^2)*y[x]^2 + 4*a^2*(b + c*K[3])*y[x]^3 + a^2*y[x]^4)), {K[3], 1, x}] + Inactive[Integrate][-((c - Sqrt[4*a^2*(1 + c^2)*(b^2 + c^2*x^2 - 2*C[1] + 2*c*x*K[4] + K[4]^2 + 2*b*(c*x + K[4]))^2 - a^4*(b^2 + c^2*x^2 - 2*C[1] + 2*c*x*K[4] + K[4]^2 + 2*b*(c*x + K[4]))^4]/(-4*(1 + c^2) + a^2*(b^2 + c^2*x^2 - 2*C[1] + 2*c*x*K[4] + K[4]^2 + 2*b*(c*x + K[4]))^2) + Inactive[Integrate][(8*a^2*c*(b + c*K[3] + K[4])*(b^2 - 2*C[1] + 2*b*(c*K[3] + K[4]) + (c*K[3] + K[4])^2))/((-4*(1 + c^2) + a^2*(b^2 - 2*C[1] + 2*b*(c*K[3] + K[4]) + (c*K[3] + K[4])^2)^2)*Sqrt[4*a^2*(1 + c^2)*(b^2 - 2*C[1] + 2*b*(c*K[3] + K[4]) + (c*K[3] + K[4])^2)^2 - a^4*(b^2 - 2*C[1] + 2*b*(c*K[3] + K[4]) + (c*K[3] + K[4])^2)^4]), {K[3], 1, x}] + c^2*Inactive[Integrate][(8*a^2*c*(b + c*K[3] + K[4])*(b^2 - 2*C[1] + 2*b*(c*K[3] + K[4]) + (c*K[3] + K[4])^2))/((-4*(1 + c^2) + a^2*(b^2 - 2*C[1] + 2*b*(c*K[3] + K[4]) + (c*K[3] + K[4])^2)^2)*Sqrt[4*a^2*(1 + c^2)*(b^2 - 2*C[1] + 2*b*(c*K[3] + K[4]) + (c*K[3] + K[4])^2)^2 - a^4*(b^2 - 2*C[1] + 2*b*(c*K[3] + K[4]) + (c*K[3] + K[4])^2)^4]), {K[3], 1, x}])/(1 + c^2)), {K[4], 1, y[x]}], y[x]]}
Maple raw input
dsolve(diff(diff(y(x),x),x) = a*(b+c*x+y(x))*(1+diff(y(x),x)^2)^(3/2), y(x))
Maple raw output
[y(x) = -c*x+RootOf(-x+Intat((4*a^2*b^2*c^2*_f^2+4*a^2*b*c^2*_f^3+a^2*c^2*_f^4-8*_C1*a^2*b*c^2*_f-4*_C1*a^2*c^2*_f^2+4*_C1^2*a^2*c^2-2*(-c^2*(_f^4*a^2+4*_f^3*a^2*b+4*_f^2*a^2*b^2-4*_C1*_f^2*a^2-8*_C1*_f*a^2*b+4*_C1^2*a^2-4*c^2-4))^(1/2)*a*b*_f-(-c^2*(_f^4*a^2+4*_f^3*a^2*b+4*_f^2*a^2*b^2-4*_C1*_f^2*a^2-8*_C1*_f*a^2*b+4*_C1^2*a^2-4*c^2-4))^(1/2)*a*_f^2-4*c^4+2*(-c^2*(_f^4*a^2+4*_f^3*a^2*b+4*_f^2*a^2*b^2-4*_C1*_f^2*a^2-8*_C1*_f*a^2*b+4*_C1^2*a^2-4*c^2-4))^(1/2)*_C1*a-4*c^2)/(_f^4*a^2+4*_f^3*a^2*b+4*_f^2*a^2*b^2-4*_C1*_f^2*a^2-8*_C1*_f*a^2*b+4*_C1^2*a^2-4*c^2-4)/(c^2+1)/c,_f = _Z)+_C2), y(x) = -c*x+RootOf(-x+Intat(-(-a^2*c^2*_f^4-4*a^2*b*c^2*_f^3-4*a^2*b^2*c^2*_f^2+4*_C1*a^2*c^2*_f^2+8*_C1*a^2*b*c^2*_f-4*_C1^2*a^2*c^2-(-c^2*(_f^4*a^2+4*_f^3*a^2*b+4*_f^2*a^2*b^2-4*_C1*_f^2*a^2-8*_C1*_f*a^2*b+4*_C1^2*a^2-4*c^2-4))^(1/2)*a*_f^2-2*(-c^2*(_f^4*a^2+4*_f^3*a^2*b+4*_f^2*a^2*b^2-4*_C1*_f^2*a^2-8*_C1*_f*a^2*b+4*_C1^2*a^2-4*c^2-4))^(1/2)*a*b*_f+4*c^4+2*(-c^2*(_f^4*a^2+4*_f^3*a^2*b+4*_f^2*a^2*b^2-4*_C1*_f^2*a^2-8*_C1*_f*a^2*b+4*_C1^2*a^2-4*c^2-4))^(1/2)*_C1*a+4*c^2)/(_f^4*a^2+4*_f^3*a^2*b+4*_f^2*a^2*b^2-4*_C1*_f^2*a^2-8*_C1*_f*a^2*b+4*_C1^2*a^2-4*c^2-4)/(c^2+1)/c,_f = _Z)+_C2)]