Internal problem ID [12128]
Internal file name [OUTPUT/10781_Tuesday_September_12_2023_08_51_50_AM_51096026/index.tex]
Book: Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS,
MOSCOW, Third printing 1977.
Section: Chapter 1, First-Order Differential Equations. Problems page 88
Problem number: Problem 18.
ODE order: 1.
ODE degree: 4.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {y-{y^{\prime }}^{4}+{y^{\prime }}^{3}=-2} \]
Integrating both sides gives \begin {align*} \int \frac {1}{\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{3}-y -2\right )}d y &= \int {dx}\\ \int _{}^{y}\frac {1}{\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{3}-\textit {\_a} -2\right )}d \textit {\_a}&= x +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {1}{\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{3}-\textit {\_a} -2\right )}d \textit {\_a} &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}\frac {1}{\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{3}-\textit {\_a} -2\right )}d \textit {\_a} = x +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y-{y^{\prime }}^{4}+{y^{\prime }}^{3}=-2 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\mathit {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{3}-y-2\right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\mathit {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{3}-y-2\right )}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\mathit {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{3}-y-2\right )}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {3 \mathit {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{3}-y-2\right )^{2}}{2}+\frac {4 \mathit {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{3}-y-2\right )^{3}}{3}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {3 c_{1} \left (\frac {\left (27+192 c_{1} +192 x +24 \sqrt {64 c_{1}^{2}+128 c_{1} x +64 x^{2}+18 c_{1} +18 x}\right )^{\frac {1}{3}}}{8}+\frac {9}{8 \left (27+192 c_{1} +192 x +24 \sqrt {64 c_{1}^{2}+128 c_{1} x +64 x^{2}+18 c_{1} +18 x}\right )^{\frac {1}{3}}}+\frac {3}{8}\right )}{4}+\frac {3 x \left (\frac {\left (27+192 c_{1} +192 x +24 \sqrt {64 c_{1}^{2}+128 c_{1} x +64 x^{2}+18 c_{1} +18 x}\right )^{\frac {1}{3}}}{8}+\frac {9}{8 \left (27+192 c_{1} +192 x +24 \sqrt {64 c_{1}^{2}+128 c_{1} x +64 x^{2}+18 c_{1} +18 x}\right )^{\frac {1}{3}}}+\frac {3}{8}\right )}{4}+\frac {9 {\left (\frac {\left (27+192 c_{1} +192 x +24 \sqrt {64 c_{1}^{2}+128 c_{1} x +64 x^{2}+18 c_{1} +18 x}\right )^{\frac {1}{3}}}{8}+\frac {9}{8 \left (27+192 c_{1} +192 x +24 \sqrt {64 c_{1}^{2}+128 c_{1} x +64 x^{2}+18 c_{1} +18 x}\right )^{\frac {1}{3}}}+\frac {3}{8}\right )}^{2}}{64}+\frac {3 c_{1}}{32}+\frac {3 x}{32}-2 \end {array} \]
Maple trace
`Methods for first order ODEs: *** Sublevel 2 *** Methods for first order ODEs: -> Solving 1st order ODE of high degree, 1st attempt trying 1st order WeierstrassP solution for high degree ODE trying 1st order WeierstrassPPrime solution for high degree ODE trying 1st order JacobiSN solution for high degree ODE trying 1st order ODE linearizable_by_differentiation trying differential order: 1; missing variables <- differential order: 1; missing x successful`
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 247
dsolve(y(x)=diff(y(x),x)^4-diff(y(x),x)^3-2,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -2 \\ y \left (x \right ) &= \frac {12 \left (\frac {243}{16384}+\frac {\left (\frac {9}{64}-c_{1} +x \right ) \sqrt {64}\, \sqrt {\left (x -c_{1} +\frac {9}{32}\right ) \left (x -c_{1} \right )}}{16}+\frac {c_{1}^{2}}{2}+\left (-\frac {9}{64}-x \right ) c_{1} +\frac {x^{2}}{2}+\frac {9 x}{64}\right ) \left (27-192 c_{1} +192 x +24 \sqrt {64}\, \sqrt {\left (x -c_{1} +\frac {9}{32}\right ) \left (x -c_{1} \right )}\right )^{\frac {2}{3}}+24 \left (x -c_{1} -\frac {7949}{1536}\right ) \left (\frac {9}{64}-c_{1} +x +\frac {\sqrt {64}\, \sqrt {\left (x -c_{1} +\frac {9}{32}\right ) \left (x -c_{1} \right )}}{8}\right ) \left (27-192 c_{1} +192 x +24 \sqrt {64}\, \sqrt {\left (x -c_{1} +\frac {9}{32}\right ) \left (x -c_{1} \right )}\right )^{\frac {1}{3}}+\frac {27 \left (\frac {9}{64}-c_{1} +x \right ) \sqrt {64}\, \sqrt {\left (x -c_{1} +\frac {9}{32}\right ) \left (x -c_{1} \right )}}{2}+108 \left (x -c_{1} +\frac {9}{128}\right ) \left (x -c_{1} +\frac {27}{128}\right )}{\left (27-192 c_{1} +192 x +24 \sqrt {64}\, \sqrt {\left (x -c_{1} +\frac {9}{32}\right ) \left (x -c_{1} \right )}\right )^{\frac {1}{3}} \left (9-64 c_{1} +64 x +8 \sqrt {64}\, \sqrt {\left (x -c_{1} +\frac {9}{32}\right ) \left (x -c_{1} \right )}\right )} \\ \end{align*}
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]==y'[x]^4-y'[x]^3-2,y[x],x,IncludeSingularSolutions -> True]
Timed out