Internal problem ID [5440]
Internal file name [OUTPUT/4931_Tuesday_February_06_2024_10_14_25_PM_21281142/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_3rd_order, _exact, _nonlinear]]
Unable to solve or complete the solution.
Unable to parse ODE.
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying 3rd order ODE linearizable_by_differentiation differential order: 3; trying a linearization to 4th order trying differential order: 3; missing variables trying differential order: 3; exact nonlinear -> Calling odsolve with the ODE`, (diff(diff(_b(_a), _a), _a))*(1+2*_b(_a)+3*_b(_a)^2)+6*_b(_a)*(diff(_b(_a), _a))^2+2*(diff(_b(_a), Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation trying 2nd order, 2 integrating factors of the form mu(x,y) trying a quadrature <- quadrature successful <- 2nd order, 2 integrating factors of the form mu(x,y) successful <- differential order: 3; exact nonlinear successful`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 1110
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 \left (x \right ) &= \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 \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \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}-432 c_{2} x^{5}+\left (1080 c_{1}^{2}+432 c_{3} +112\right ) x^{4}+5184 c_{1} c_{2} x^{3}+\left (2592 c_{1}^{3}+\left (-5184 c_{3} -1344\right ) c_{1} +5184 c_{2}^{2}\right ) x^{2}+5184 c_{2} \left (c_{1}^{2}-2 c_{3} -\frac {14}{27}\right ) x +1296 c_{1}^{4}+\left (-5184 c_{3} -1344\right ) c_{1}^{2}+5184 c_{3}^{2}+2688 c_{3} +576}\right )^{\frac {2}{3}}+32 i \sqrt {3}+8 \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}-432 c_{2} x^{5}+\left (1080 c_{1}^{2}+432 c_{3} +112\right ) x^{4}+5184 c_{1} c_{2} x^{3}+\left (2592 c_{1}^{3}+\left (-5184 c_{3} -1344\right ) c_{1} +5184 c_{2}^{2}\right ) x^{2}+5184 c_{2} \left (c_{1}^{2}-2 c_{3} -\frac {14}{27}\right ) x +1296 c_{1}^{4}+\left (-5184 c_{3} -1344\right ) c_{1}^{2}+5184 c_{3}^{2}+2688 c_{3} +576}\right )^{\frac {1}{3}}-32}{24 \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}-432 c_{2} x^{5}+\left (1080 c_{1}^{2}+432 c_{3} +112\right ) x^{4}+5184 c_{1} c_{2} x^{3}+\left (2592 c_{1}^{3}+\left (-5184 c_{3} -1344\right ) c_{1} +5184 c_{2}^{2}\right ) x^{2}+5184 c_{2} \left (c_{1}^{2}-2 c_{3} -\frac {14}{27}\right ) x +1296 c_{1}^{4}+\left (-5184 c_{3} -1344\right ) c_{1}^{2}+5184 c_{3}^{2}+2688 c_{3} +576}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \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}-432 c_{2} x^{5}+\left (1080 c_{1}^{2}+432 c_{3} +112\right ) x^{4}+5184 c_{1} c_{2} x^{3}+\left (2592 c_{1}^{3}+\left (-5184 c_{3} -1344\right ) c_{1} +5184 c_{2}^{2}\right ) x^{2}+5184 c_{2} \left (c_{1}^{2}-2 c_{3} -\frac {14}{27}\right ) x +1296 c_{1}^{4}+\left (-5184 c_{3} -1344\right ) c_{1}^{2}+5184 c_{3}^{2}+2688 c_{3} +576}\right )^{\frac {2}{3}}+32 i \sqrt {3}-8 \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}-432 c_{2} x^{5}+\left (1080 c_{1}^{2}+432 c_{3} +112\right ) x^{4}+5184 c_{1} c_{2} x^{3}+\left (2592 c_{1}^{3}+\left (-5184 c_{3} -1344\right ) c_{1} +5184 c_{2}^{2}\right ) x^{2}+5184 c_{2} \left (c_{1}^{2}-2 c_{3} -\frac {14}{27}\right ) x +1296 c_{1}^{4}+\left (-5184 c_{3} -1344\right ) c_{1}^{2}+5184 c_{3}^{2}+2688 c_{3} +576}\right )^{\frac {1}{3}}+32}{24 \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}-432 c_{2} x^{5}+\left (1080 c_{1}^{2}+432 c_{3} +112\right ) x^{4}+5184 c_{1} c_{2} x^{3}+\left (2592 c_{1}^{3}+\left (-5184 c_{3} -1344\right ) c_{1} +5184 c_{2}^{2}\right ) x^{2}+5184 c_{2} \left (c_{1}^{2}-2 c_{3} -\frac {14}{27}\right ) x +1296 c_{1}^{4}+\left (-5184 c_{3} -1344\right ) c_{1}^{2}+5184 c_{3}^{2}+2688 c_{3} +576}\right )^{\frac {1}{3}}} \\ \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*}