Internal problem ID [5440]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 19. Linear equations with variable coefficients (Misc. types). Supplemetary
problems. Page 132
Problem number: 33.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _exact, _nonlinear]]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 1500
dsolve((1+2*y(x)+3*y(x)^2)*diff(y(x),x$3)+6*diff(y(x),x)*( diff(y(x),x$2)+diff(y(x),x)^2+3*y(x)*diff(y(x),x$2) )=x,y(x), singsol=all)
\begin{align*} y = \frac {\left (224+36 x^{4}-432 c_{1} x^{2}-432 c_{1}^{2}-864 c_{2} x +864 c_{3} +12 \sqrt {9 x^{8}-216 c_{1} x^{6}+1080 c_{1}^{2} x^{4}-432 c_{2} x^{5}+2592 c_{1}^{3} x^{2}+5184 c_{1} c_{2} x^{3}+432 c_{3} x^{4}+1296 c_{1}^{4}+5184 c_{1}^{2} c_{2} x -5184 c_{1} c_{3} x^{2}+5184 c_{2}^{2} x^{2}+112 x^{4}-5184 c_{1}^{2} c_{3} -1344 c_{1} x^{2}-10368 c_{2} c_{3} x -1344 c_{1}^{2}-2688 c_{2} x +5184 c_{3}^{2}+2688 c_{3} +576}\right )^{\frac {1}{3}}}{12}-\frac {8}{3 \left (224+36 x^{4}-432 c_{1} x^{2}-432 c_{1}^{2}-864 c_{2} x +864 c_{3} +12 \sqrt {9 x^{8}-216 c_{1} x^{6}+1080 c_{1}^{2} x^{4}-432 c_{2} x^{5}+2592 c_{1}^{3} x^{2}+5184 c_{1} c_{2} x^{3}+432 c_{3} x^{4}+1296 c_{1}^{4}+5184 c_{1}^{2} c_{2} x -5184 c_{1} c_{3} x^{2}+5184 c_{2}^{2} x^{2}+112 x^{4}-5184 c_{1}^{2} c_{3} -1344 c_{1} x^{2}-10368 c_{2} c_{3} x -1344 c_{1}^{2}-2688 c_{2} x +5184 c_{3}^{2}+2688 c_{3} +576}\right )^{\frac {1}{3}}}-\frac {1}{3} y = -\frac {\left (224+36 x^{4}-432 c_{1} x^{2}-432 c_{1}^{2}-864 c_{2} x +864 c_{3} +12 \sqrt {9 x^{8}-216 c_{1} x^{6}+1080 c_{1}^{2} x^{4}-432 c_{2} x^{5}+2592 c_{1}^{3} x^{2}+5184 c_{1} c_{2} x^{3}+432 c_{3} x^{4}+1296 c_{1}^{4}+5184 c_{1}^{2} c_{2} x -5184 c_{1} c_{3} x^{2}+5184 c_{2}^{2} x^{2}+112 x^{4}-5184 c_{1}^{2} c_{3} -1344 c_{1} x^{2}-10368 c_{2} c_{3} x -1344 c_{1}^{2}-2688 c_{2} x +5184 c_{3}^{2}+2688 c_{3} +576}\right )^{\frac {1}{3}}}{24}+\frac {4}{3 \left (224+36 x^{4}-432 c_{1} x^{2}-432 c_{1}^{2}-864 c_{2} x +864 c_{3} +12 \sqrt {9 x^{8}-216 c_{1} x^{6}+1080 c_{1}^{2} x^{4}-432 c_{2} x^{5}+2592 c_{1}^{3} x^{2}+5184 c_{1} c_{2} x^{3}+432 c_{3} x^{4}+1296 c_{1}^{4}+5184 c_{1}^{2} c_{2} x -5184 c_{1} c_{3} x^{2}+5184 c_{2}^{2} x^{2}+112 x^{4}-5184 c_{1}^{2} c_{3} -1344 c_{1} x^{2}-10368 c_{2} c_{3} x -1344 c_{1}^{2}-2688 c_{2} x +5184 c_{3}^{2}+2688 c_{3} +576}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (224+36 x^{4}-432 c_{1} x^{2}-432 c_{1}^{2}-864 c_{2} x +864 c_{3} +12 \sqrt {9 x^{8}-216 c_{1} x^{6}+1080 c_{1}^{2} x^{4}-432 c_{2} x^{5}+2592 c_{1}^{3} x^{2}+5184 c_{1} c_{2} x^{3}+432 c_{3} x^{4}+1296 c_{1}^{4}+5184 c_{1}^{2} c_{2} x -5184 c_{1} c_{3} x^{2}+5184 c_{2}^{2} x^{2}+112 x^{4}-5184 c_{1}^{2} c_{3} -1344 c_{1} x^{2}-10368 c_{2} c_{3} x -1344 c_{1}^{2}-2688 c_{2} x +5184 c_{3}^{2}+2688 c_{3} +576}\right )^{\frac {1}{3}}}{6}+\frac {16}{3 \left (224+36 x^{4}-432 c_{1} x^{2}-432 c_{1}^{2}-864 c_{2} x +864 c_{3} +12 \sqrt {9 x^{8}-216 c_{1} x^{6}+1080 c_{1}^{2} x^{4}-432 c_{2} x^{5}+2592 c_{1}^{3} x^{2}+5184 c_{1} c_{2} x^{3}+432 c_{3} x^{4}+1296 c_{1}^{4}+5184 c_{1}^{2} c_{2} x -5184 c_{1} c_{3} x^{2}+5184 c_{2}^{2} x^{2}+112 x^{4}-5184 c_{1}^{2} c_{3} -1344 c_{1} x^{2}-10368 c_{2} c_{3} x -1344 c_{1}^{2}-2688 c_{2} x +5184 c_{3}^{2}+2688 c_{3} +576}\right )^{\frac {1}{3}}}\right )}{4} y = -\frac {\left (224+36 x^{4}-432 c_{1} x^{2}-432 c_{1}^{2}-864 c_{2} x +864 c_{3} +12 \sqrt {9 x^{8}-216 c_{1} x^{6}+1080 c_{1}^{2} x^{4}-432 c_{2} x^{5}+2592 c_{1}^{3} x^{2}+5184 c_{1} c_{2} x^{3}+432 c_{3} x^{4}+1296 c_{1}^{4}+5184 c_{1}^{2} c_{2} x -5184 c_{1} c_{3} x^{2}+5184 c_{2}^{2} x^{2}+112 x^{4}-5184 c_{1}^{2} c_{3} -1344 c_{1} x^{2}-10368 c_{2} c_{3} x -1344 c_{1}^{2}-2688 c_{2} x +5184 c_{3}^{2}+2688 c_{3} +576}\right )^{\frac {1}{3}}}{24}+\frac {4}{3 \left (224+36 x^{4}-432 c_{1} x^{2}-432 c_{1}^{2}-864 c_{2} x +864 c_{3} +12 \sqrt {9 x^{8}-216 c_{1} x^{6}+1080 c_{1}^{2} x^{4}-432 c_{2} x^{5}+2592 c_{1}^{3} x^{2}+5184 c_{1} c_{2} x^{3}+432 c_{3} x^{4}+1296 c_{1}^{4}+5184 c_{1}^{2} c_{2} x -5184 c_{1} c_{3} x^{2}+5184 c_{2}^{2} x^{2}+112 x^{4}-5184 c_{1}^{2} c_{3} -1344 c_{1} x^{2}-10368 c_{2} c_{3} x -1344 c_{1}^{2}-2688 c_{2} x +5184 c_{3}^{2}+2688 c_{3} +576}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (224+36 x^{4}-432 c_{1} x^{2}-432 c_{1}^{2}-864 c_{2} x +864 c_{3} +12 \sqrt {9 x^{8}-216 c_{1} x^{6}+1080 c_{1}^{2} x^{4}-432 c_{2} x^{5}+2592 c_{1}^{3} x^{2}+5184 c_{1} c_{2} x^{3}+432 c_{3} x^{4}+1296 c_{1}^{4}+5184 c_{1}^{2} c_{2} x -5184 c_{1} c_{3} x^{2}+5184 c_{2}^{2} x^{2}+112 x^{4}-5184 c_{1}^{2} c_{3} -1344 c_{1} x^{2}-10368 c_{2} c_{3} x -1344 c_{1}^{2}-2688 c_{2} x +5184 c_{3}^{2}+2688 c_{3} +576}\right )^{\frac {1}{3}}}{6}+\frac {16}{3 \left (224+36 x^{4}-432 c_{1} x^{2}-432 c_{1}^{2}-864 c_{2} x +864 c_{3} +12 \sqrt {9 x^{8}-216 c_{1} x^{6}+1080 c_{1}^{2} x^{4}-432 c_{2} x^{5}+2592 c_{1}^{3} x^{2}+5184 c_{1} c_{2} x^{3}+432 c_{3} x^{4}+1296 c_{1}^{4}+5184 c_{1}^{2} c_{2} x -5184 c_{1} c_{3} x^{2}+5184 c_{2}^{2} x^{2}+112 x^{4}-5184 c_{1}^{2} c_{3} -1344 c_{1} x^{2}-10368 c_{2} c_{3} x -1344 c_{1}^{2}-2688 c_{2} x +5184 c_{3}^{2}+2688 c_{3} +576}\right )^{\frac {1}{3}}}\right )}{4} \end{align*}
✓ Solution by Mathematica
Time used: 0.369 (sec). Leaf size: 523
DSolve[(1+2*y[x]+3*y[x]^2)*y'''[x]+6*y'[x]*( y''[x]+y'[x]^2+3*y[x]*y''[x] )==x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2^{2/3} \left (9 x^4+108 c_1 x^2+\sqrt {2048+\left (9 x^4+108 c_1 x^2+27 c_3 x+56+216 c_2\right ){}^2}+27 c_3 x+56+216 c_2\right ){}^{2/3}-4 \sqrt [3]{9 x^4+108 c_1 x^2+\sqrt {2048+\left (9 x^4+108 c_1 x^2+27 c_3 x+56+216 c_2\right ){}^2}+27 c_3 x+56+216 c_2}-16 \sqrt [3]{2}}{12 \sqrt [3]{9 x^4+108 c_1 x^2+\sqrt {2048+\left (9 x^4+108 c_1 x^2+27 c_3 x+56+216 c_2\right ){}^2}+27 c_3 x+56+216 c_2}} y(x)\to \frac {1}{24} \left (i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{9 x^4+108 c_1 x^2+\sqrt {2048+\left (9 x^4+108 c_1 x^2+27 c_3 x+56+216 c_2\right ){}^2}+27 c_3 x+56+216 c_2}+\frac {16 \sqrt [3]{2} \left (1+i \sqrt {3}\right )}{\sqrt [3]{9 x^4+108 c_1 x^2+\sqrt {2048+\left (9 x^4+108 c_1 x^2+27 c_3 x+56+216 c_2\right ){}^2}+27 c_3 x+56+216 c_2}}-8\right ) y(x)\to \frac {1}{24} \left (-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{9 x^4+108 c_1 x^2+\sqrt {2048+\left (9 x^4+108 c_1 x^2+27 c_3 x+56+216 c_2\right ){}^2}+27 c_3 x+56+216 c_2}+\frac {16 \sqrt [3]{2} \left (1-i \sqrt {3}\right )}{\sqrt [3]{9 x^4+108 c_1 x^2+\sqrt {2048+\left (9 x^4+108 c_1 x^2+27 c_3 x+56+216 c_2\right ){}^2}+27 c_3 x+56+216 c_2}}-8\right ) \end{align*}