Internal problem ID [10126]
Internal file name [OUTPUT/9073_Monday_June_06_2022_06_22_06_AM_68206313/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1805 (book 6.214).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_ode_missing_x"
Maple gives the following as the ode type
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
\[ \boxed {\left (4 y^{3}-a y-b \right ) y^{\prime \prime }-\left (6 y^{2}-\frac {a}{2}\right ) {y^{\prime }}^{2}=0} \]
This is missing independent variable second order ode. Solved by reduction of order by using substitution which makes the dependent variable \(y\) an independent variable. Using \begin {align*} y' &= p(y) \end {align*}
Then \begin {align*} y'' &= \frac {dp}{dx}\\ &= \frac {dy}{dx} \frac {dp}{dy}\\ &= p \frac {dp}{dy} \end {align*}
Hence the ode becomes \begin {align*} \left (4 y^{3}-y a -b \right ) p \left (y \right ) \left (\frac {d}{d y}p \left (y \right )\right )+\left (-6 p \left (y \right ) y^{2}+\frac {a p \left (y \right )}{2}\right ) p \left (y \right ) = 0 \end {align*}
Which is now solved as first order ode for \(p(y)\). In canonical form the ODE is \begin {align*} p' &= F(y,p)\\ &= f( y) g(p)\\ &= \frac {p \left (-12 y^{2}+a \right )}{-8 y^{3}+2 y a +2 b} \end {align*}
Where \(f(y)=\frac {-12 y^{2}+a}{-8 y^{3}+2 y a +2 b}\) and \(g(p)=p\). Integrating both sides gives \begin {align*} \frac {1}{p} \,dp &= \frac {-12 y^{2}+a}{-8 y^{3}+2 y a +2 b} \,d y\\ \int { \frac {1}{p} \,dp} &= \int {\frac {-12 y^{2}+a}{-8 y^{3}+2 y a +2 b} \,d y}\\ \ln \left (p \right )&=\frac {\ln \left (4 y^{3}-y a -b \right )}{2}+c_{1}\\ p&={\mathrm e}^{\frac {\ln \left (4 y^{3}-y a -b \right )}{2}+c_{1}}\\ &=c_{1} \sqrt {4 y^{3}-y a -b} \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*} y^{\prime } = c_{1} \sqrt {4 y^{3}-a y-b} \end {align*}
Integrating both sides gives \begin {align*} \int \frac {1}{c_{1} \sqrt {4 y^{3}-y a -b}}d y &= \int {dx}\\ \int _{}^{y}\frac {1}{c_{1} \sqrt {4 \textit {\_a}^{3}-\textit {\_a} a -b}}d \textit {\_a}&= x +c_{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {1}{c_{1} \sqrt {4 \textit {\_a}^{3}-\textit {\_a} a -b}}d \textit {\_a} &= x +c_{2} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}\frac {1}{c_{1} \sqrt {4 \textit {\_a}^{3}-\textit {\_a} a -b}}d \textit {\_a} = x +c_{2} \] Verified OK.
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville <- 2nd_order Liouville successful`
✓ Solution by Maple
Time used: 19.813 (sec). Leaf size: 254
dsolve((4*y(x)^3-a*y(x)-b)*diff(diff(y(x),x),x)-(6*y(x)^2-1/2*a)*diff(y(x),x)^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}+3 a}{6 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {-i \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}} \sqrt {3}+3 i \sqrt {3}\, a -\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}-3 a}{12 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {-i \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}} \sqrt {3}+3 i \sqrt {3}\, a +\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}+3 a}{12 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}} \\ \int _{}^{y \left (x \right )}\frac {1}{\sqrt {4 \textit {\_a}^{3}-a \textit {\_a} -b}}d \textit {\_a} -c_{1} x -c_{2} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 12.733 (sec). Leaf size: 416
DSolve[(a/2 - 6*y[x]^2)*y'[x]^2 + (-b - a*y[x] + 4*y[x]^3)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\sqrt {2} \sqrt {\frac {y(x)-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,1\right ]}{\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,1\right ]}} \sqrt {\frac {y(x)-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,2\right ]}{\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,2\right ]}} \left (y(x)-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,3\right ]-y(x)}{\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,3\right ]-\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,2\right ]}}\right ),\frac {\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,2\right ]-\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,3\right ]}{\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,1\right ]-\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,3\right ]}\right )}{c_1 \sqrt {a y(x)+b-4 y(x)^3} \sqrt {\frac {y(x)-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]}{\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,2\right ]-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]}}}=x+c_2,y(x)\right ] \]