Internal problem ID [13839]
Internal file name [OUTPUT/13011_Friday_February_23_2024_06_46_23_AM_96052526/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 25. Review exercises for part III. page 447
Problem number: 42.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_ode_missing_y"
Maple gives the following as the ode type
[[_2nd_order, _missing_y]]
\[ \boxed {x y^{\prime \prime }-y^{\prime }+3 x {y^{\prime }}^{3}=0} \]
This is second order ode with missing dependent variable \(y\). Let \begin {align*} p(x) &= y^{\prime } \end {align*}
Then \begin {align*} p'(x) &= y^{\prime \prime } \end {align*}
Hence the ode becomes \begin {align*} p^{\prime }\left (x \right ) x +\left (-1+3 x p \left (x \right )^{2}\right ) p \left (x \right ) = 0 \end {align*}
Which is now solve for \(p(x)\) as first order ode. Using the change of variables \(p \left (x \right ) = u \left (x \right ) x\) on the above ode results in new ode in \(u \left (x \right )\) \begin {align*} \left (u^{\prime }\left (x \right ) x +u \left (x \right )\right ) x +\left (-1+3 x^{3} u \left (x \right )^{2}\right ) u \left (x \right ) x = 0 \end {align*}
In canonical form the ODE is \begin {align*} u' &= F(x,u)\\ &= f( x) g(u)\\ &= -3 u^{3} x^{2} \end {align*}
Where \(f(x)=-3 x^{2}\) and \(g(u)=u^{3}\). Integrating both sides gives \begin{align*} \frac {1}{u^{3}} \,du &= -3 x^{2} \,d x \\ \int { \frac {1}{u^{3}} \,du} &= \int {-3 x^{2} \,d x} \\ -\frac {1}{2 u^{2}}&=-x^{3}+c_{2} \\ \end{align*} The solution is \[ -\frac {1}{2 u \left (x \right )^{2}}+x^{3}-c_{2} = 0 \] Replacing \(u(x)\) in the above solution by \(\frac {p \left (x \right )}{x}\) results in the solution for \(p \left (x \right )\) in implicit form \begin {align*} -\frac {x^{2}}{2 p \left (x \right )^{2}}+x^{3}-c_{2} = 0\\ -\frac {x^{2}}{2 p \left (x \right )^{2}}+x^{3}-c_{2} = 0 \end {align*}
For solution (1) found earlier, since \(p=y^{\prime }\) then we now have a new first order ode to solve which is \begin {align*} -\frac {x^{2}}{2 {y^{\prime }}^{2}}+x^{3}-c_{2} = 0 \end {align*}
Solving the given ode for \(y^{\prime }\) results in \(2\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} y^{\prime }&=-\frac {x}{\sqrt {2 x^{3}-2 c_{2}}} \tag {1} \\ y^{\prime }&=\frac {x}{\sqrt {2 x^{3}-2 c_{2}}} \tag {2} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Integrating both sides gives \begin {align*} y &= \int { -\frac {x}{\sqrt {2 x^{3}-2 c_{2}}}\,\mathop {\mathrm {d}x}}\\ &= -\frac {2 i \sqrt {3}\, c_{2}^{{1}/{3}} \sqrt {-\frac {i \left (x +\frac {c_{2}^{{1}/{3}}}{2}+\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \sqrt {3}}{c_{2}^{{1}/{3}}}}\, \sqrt {\frac {x -c_{2}^{{1}/{3}}}{-\frac {3 c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}}}\, \sqrt {\frac {i \left (x +\frac {c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \sqrt {3}}{c_{2}^{{1}/{3}}}}\, \left (\left (-\frac {3 c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {c_{2}^{{1}/{3}}}{2}+\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \sqrt {3}}{c_{2}^{{1}/{3}}}}}{3}, \sqrt {-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{-\frac {3 c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}}}\right )+c_{2}^{{1}/{3}} \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {c_{2}^{{1}/{3}}}{2}+\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \sqrt {3}}{c_{2}^{{1}/{3}}}}}{3}, \sqrt {-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{-\frac {3 c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}}}\right )\right )}{3 \sqrt {2 x^{3}-2 c_{2}}}+c_{3} \end {align*}
Solving equation (2)
Integrating both sides gives \begin {align*} y &= \int { \frac {x}{\sqrt {2 x^{3}-2 c_{2}}}\,\mathop {\mathrm {d}x}}\\ &= \frac {2 i \sqrt {3}\, c_{2}^{{1}/{3}} \sqrt {-\frac {i \left (x +\frac {c_{2}^{{1}/{3}}}{2}+\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \sqrt {3}}{c_{2}^{{1}/{3}}}}\, \sqrt {\frac {x -c_{2}^{{1}/{3}}}{-\frac {3 c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}}}\, \sqrt {\frac {i \left (x +\frac {c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \sqrt {3}}{c_{2}^{{1}/{3}}}}\, \left (\left (-\frac {3 c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {c_{2}^{{1}/{3}}}{2}+\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \sqrt {3}}{c_{2}^{{1}/{3}}}}}{3}, \sqrt {-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{-\frac {3 c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}}}\right )+c_{2}^{{1}/{3}} \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {c_{2}^{{1}/{3}}}{2}+\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \sqrt {3}}{c_{2}^{{1}/{3}}}}}{3}, \sqrt {-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{-\frac {3 c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}}}\right )\right )}{3 \sqrt {2 x^{3}-2 c_{2}}}+c_{4} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {2 i \sqrt {3}\, c_{2}^{{1}/{3}} \sqrt {-\frac {i \left (x +\frac {c_{2}^{{1}/{3}}}{2}+\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \sqrt {3}}{c_{2}^{{1}/{3}}}}\, \sqrt {\frac {x -c_{2}^{{1}/{3}}}{-\frac {3 c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}}}\, \sqrt {\frac {i \left (x +\frac {c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \sqrt {3}}{c_{2}^{{1}/{3}}}}\, \left (\left (-\frac {3 c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {c_{2}^{{1}/{3}}}{2}+\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \sqrt {3}}{c_{2}^{{1}/{3}}}}}{3}, \sqrt {-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{-\frac {3 c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}}}\right )+c_{2}^{{1}/{3}} \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {c_{2}^{{1}/{3}}}{2}+\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \sqrt {3}}{c_{2}^{{1}/{3}}}}}{3}, \sqrt {-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{-\frac {3 c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}}}\right )\right )}{3 \sqrt {2 x^{3}-2 c_{2}}}+c_{3} \\ \tag{2} y &= \frac {2 i \sqrt {3}\, c_{2}^{{1}/{3}} \sqrt {-\frac {i \left (x +\frac {c_{2}^{{1}/{3}}}{2}+\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \sqrt {3}}{c_{2}^{{1}/{3}}}}\, \sqrt {\frac {x -c_{2}^{{1}/{3}}}{-\frac {3 c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}}}\, \sqrt {\frac {i \left (x +\frac {c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \sqrt {3}}{c_{2}^{{1}/{3}}}}\, \left (\left (-\frac {3 c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {c_{2}^{{1}/{3}}}{2}+\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \sqrt {3}}{c_{2}^{{1}/{3}}}}}{3}, \sqrt {-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{-\frac {3 c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}}}\right )+c_{2}^{{1}/{3}} \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {c_{2}^{{1}/{3}}}{2}+\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \sqrt {3}}{c_{2}^{{1}/{3}}}}}{3}, \sqrt {-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{-\frac {3 c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}}}\right )\right )}{3 \sqrt {2 x^{3}-2 c_{2}}}+c_{4} \\ \end{align*}
Verification of solutions
\[ y = -\frac {2 i \sqrt {3}\, c_{2}^{{1}/{3}} \sqrt {-\frac {i \left (x +\frac {c_{2}^{{1}/{3}}}{2}+\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \sqrt {3}}{c_{2}^{{1}/{3}}}}\, \sqrt {\frac {x -c_{2}^{{1}/{3}}}{-\frac {3 c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}}}\, \sqrt {\frac {i \left (x +\frac {c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \sqrt {3}}{c_{2}^{{1}/{3}}}}\, \left (\left (-\frac {3 c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {c_{2}^{{1}/{3}}}{2}+\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \sqrt {3}}{c_{2}^{{1}/{3}}}}}{3}, \sqrt {-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{-\frac {3 c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}}}\right )+c_{2}^{{1}/{3}} \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {c_{2}^{{1}/{3}}}{2}+\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \sqrt {3}}{c_{2}^{{1}/{3}}}}}{3}, \sqrt {-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{-\frac {3 c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}}}\right )\right )}{3 \sqrt {2 x^{3}-2 c_{2}}}+c_{3} \] Verified OK.
\[ y = \frac {2 i \sqrt {3}\, c_{2}^{{1}/{3}} \sqrt {-\frac {i \left (x +\frac {c_{2}^{{1}/{3}}}{2}+\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \sqrt {3}}{c_{2}^{{1}/{3}}}}\, \sqrt {\frac {x -c_{2}^{{1}/{3}}}{-\frac {3 c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}}}\, \sqrt {\frac {i \left (x +\frac {c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \sqrt {3}}{c_{2}^{{1}/{3}}}}\, \left (\left (-\frac {3 c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {c_{2}^{{1}/{3}}}{2}+\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \sqrt {3}}{c_{2}^{{1}/{3}}}}}{3}, \sqrt {-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{-\frac {3 c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}}}\right )+c_{2}^{{1}/{3}} \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {c_{2}^{{1}/{3}}}{2}+\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}\right ) \sqrt {3}}{c_{2}^{{1}/{3}}}}}{3}, \sqrt {-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{-\frac {3 c_{2}^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, c_{2}^{{1}/{3}}}{2}}}\right )\right )}{3 \sqrt {2 x^{3}-2 c_{2}}}+c_{4} \] Verified OK.
Maple trace
`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 a change of variables {x -> y(x), y(x) -> x} 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 differential order: 2; missing variables `, `-> Computing symmetries using: way = 3 -> Calling odsolve with the ODE`, diff(_b(_a), _a) = -_b(_a)*(3*_a*_b(_a)^2-1)/_a, _b(_a), HINT = [[_a, -(1/2)*_b]]` *** Sublevel symmetry methods on request `, `1st order, trying reduction of order with given symmetries:`[_a, -1/2*_b]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 43
dsolve(x*diff(y(x),x$2)-diff(y(x),x)=-3*x* diff(y(x),x)^3,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \int \frac {x}{\sqrt {2 x^{3}-c_{1}}}d x +c_{2} \\ y \left (x \right ) &= -\left (\int \frac {x}{\sqrt {2 x^{3}-c_{1}}}d x \right )+c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.949 (sec). Leaf size: 195
DSolve[x*y''[x]-y'[x]==-3*x* (y'[x])^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2-\frac {x^2 \sqrt {1+\frac {2 x^3}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},-\frac {2 x^3}{c_1}\right )}{2 \sqrt {2 x^3+c_1}} \\ y(x)\to \frac {x^2 \sqrt {1+\frac {2 x^3}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},-\frac {2 x^3}{c_1}\right )}{2 \sqrt {2 x^3+c_1}}+c_2 \\ y(x)\to c_2 \\ y(x)\to -\frac {3 \sqrt {x^3} \operatorname {Gamma}\left (\frac {5}{3}\right )}{\sqrt {2} x \operatorname {Gamma}\left (\frac {2}{3}\right )}+c_2 \\ y(x)\to \frac {3 \sqrt {x^3} \operatorname {Gamma}\left (\frac {5}{3}\right )}{\sqrt {2} x \operatorname {Gamma}\left (\frac {2}{3}\right )}+c_2 \\ \end{align*}