Internal problem ID [10148]
Internal file name [OUTPUT/9095_Monday_June_06_2022_06_32_54_AM_88049679/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1827 (book 6.236).
ODE order: 2.
ODE degree: 2.
The type(s) of ODE detected by this program : "second_order_ode_high_degree"
Maple gives the following as the ode type
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\[ \boxed {{y^{\prime \prime }}^{2}-a y=b} \] Does not support ODE with \({y^{\prime \prime }}^{n}\) where \(n\neq 1\) unless \(-a\) is missing which is not the case here.
Solving for \(y^{\prime \prime }\) from the ode gives \begin{align*} \tag{1} y^{\prime \prime } &= \sqrt {a y+b} \\ \tag{2} y^{\prime \prime } &= -\sqrt {a y+b} \\ \end{align*} Now each ode is solved. Multiplying the ode by \(y^{\prime }\) gives \[ y^{\prime } y^{\prime \prime }-y^{\prime } \sqrt {a y+b} = 0 \] Integrating the above w.r.t \(x\) gives \begin {align*} \int \left (y^{\prime } y^{\prime \prime }-y^{\prime } \sqrt {a y+b}\right )d x &= 0 \\ \frac {{y^{\prime }}^{2}}{2}-\frac {2 \left (a y+b \right )^{\frac {3}{2}}}{3 a} = c_2 \end {align*}
Which is now solved for \(y\). 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 {\sqrt {6}\, \sqrt {a \left (2 \sqrt {a y+b}\, a y+2 \sqrt {a y+b}\, b +3 c_{1} a \right )}}{3 a} \tag {1} \\ y^{\prime }&=-\frac {\sqrt {6}\, \sqrt {a \left (2 \sqrt {a y+b}\, a y+2 \sqrt {a y+b}\, b +3 c_{1} a \right )}}{3 a} \tag {2} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Integrating both sides gives \begin {align*} \int \frac {a \sqrt {6}}{2 \sqrt {a \left (2 \sqrt {y a +b}\, a y +2 \sqrt {y a +b}\, b +3 c_{1} a \right )}}d y &= \int {dx}\\ \int _{}^{y}\frac {a \sqrt {6}}{2 \sqrt {a \left (2 \sqrt {\textit {\_a} a +b}\, a \textit {\_a} +2 \sqrt {\textit {\_a} a +b}\, b +3 c_{1} a \right )}}d \textit {\_a}&= x +c_{2} \end {align*}
Solving equation (2)
Integrating both sides gives \begin {align*} \int -\frac {a \sqrt {6}}{2 \sqrt {a \left (2 \sqrt {y a +b}\, a y +2 \sqrt {y a +b}\, b +3 c_{1} a \right )}}d y &= \int {dx}\\ \int _{}^{y}-\frac {a \sqrt {6}}{2 \sqrt {a \left (2 \sqrt {\textit {\_a} a +b}\, a \textit {\_a} +2 \sqrt {\textit {\_a} a +b}\, b +3 c_{1} a \right )}}d \textit {\_a}&= c_{3} +x \end {align*}
Multiplying the ode by \(y^{\prime }\) gives \[ y^{\prime } y^{\prime \prime }+y^{\prime } \sqrt {a y+b} = 0 \] Integrating the above w.r.t \(x\) gives \begin {align*} \int \left (y^{\prime } y^{\prime \prime }+y^{\prime } \sqrt {a y+b}\right )d x &= 0 \\ \frac {2 \left (a y+b \right )^{\frac {3}{2}}}{3 a}+\frac {{y^{\prime }}^{2}}{2} = c_2 \end {align*}
Which is now solved for \(y\). 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 {\sqrt {-6 a \left (2 \sqrt {a y+b}\, a y+2 \sqrt {a y+b}\, b -3 c_{4} a \right )}}{3 a} \tag {1} \\ y^{\prime }&=-\frac {\sqrt {-6 a \left (2 \sqrt {a y+b}\, a y+2 \sqrt {a y+b}\, b -3 c_{4} a \right )}}{3 a} \tag {2} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Integrating both sides gives \begin {align*} \int \frac {3 a}{\sqrt {-6 a \left (2 \sqrt {y a +b}\, a y +2 \sqrt {y a +b}\, b -3 c_{4} a \right )}}d y &= \int {dx}\\ \int _{}^{y}\frac {3 a}{\sqrt {-6 a \left (2 \sqrt {\textit {\_a} a +b}\, a \textit {\_a} +2 \sqrt {\textit {\_a} a +b}\, b -3 c_{4} a \right )}}d \textit {\_a}&= x +c_{5} \end {align*}
Solving equation (2)
Integrating both sides gives \begin {align*} \int -\frac {3 a}{\sqrt {-6 a \left (2 \sqrt {y a +b}\, a y +2 \sqrt {y a +b}\, b -3 c_{4} a \right )}}d y &= \int {dx}\\ \int _{}^{y}-\frac {3 a}{\sqrt {-6 a \left (2 \sqrt {\textit {\_a} a +b}\, a \textit {\_a} +2 \sqrt {\textit {\_a} a +b}\, b -3 c_{4} a \right )}}d \textit {\_a}&= x +c_{6} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {a \sqrt {6}}{2 \sqrt {a \left (2 \sqrt {\textit {\_a} a +b}\, a \textit {\_a} +2 \sqrt {\textit {\_a} a +b}\, b +3 c_{1} a \right )}}d \textit {\_a} &= x +c_{2} \\ \tag{2} \int _{}^{y}-\frac {a \sqrt {6}}{2 \sqrt {a \left (2 \sqrt {\textit {\_a} a +b}\, a \textit {\_a} +2 \sqrt {\textit {\_a} a +b}\, b +3 c_{1} a \right )}}d \textit {\_a} &= c_{3} +x \\ \tag{3} \int _{}^{y}\frac {3 a}{\sqrt {-6 a \left (2 \sqrt {\textit {\_a} a +b}\, a \textit {\_a} +2 \sqrt {\textit {\_a} a +b}\, b -3 c_{4} a \right )}}d \textit {\_a} &= x +c_{5} \\ \tag{4} \int _{}^{y}-\frac {3 a}{\sqrt {-6 a \left (2 \sqrt {\textit {\_a} a +b}\, a \textit {\_a} +2 \sqrt {\textit {\_a} a +b}\, b -3 c_{4} a \right )}}d \textit {\_a} &= x +c_{6} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}\frac {a \sqrt {6}}{2 \sqrt {a \left (2 \sqrt {\textit {\_a} a +b}\, a \textit {\_a} +2 \sqrt {\textit {\_a} a +b}\, b +3 c_{1} a \right )}}d \textit {\_a} = x +c_{2} \] Verified OK.
\[ \int _{}^{y}-\frac {a \sqrt {6}}{2 \sqrt {a \left (2 \sqrt {\textit {\_a} a +b}\, a \textit {\_a} +2 \sqrt {\textit {\_a} a +b}\, b +3 c_{1} a \right )}}d \textit {\_a} = c_{3} +x \] Verified OK.
\[ \int _{}^{y}\frac {3 a}{\sqrt {-6 a \left (2 \sqrt {\textit {\_a} a +b}\, a \textit {\_a} +2 \sqrt {\textit {\_a} a +b}\, b -3 c_{4} a \right )}}d \textit {\_a} = x +c_{5} \] Verified OK.
\[ \int _{}^{y}-\frac {3 a}{\sqrt {-6 a \left (2 \sqrt {\textit {\_a} a +b}\, a \textit {\_a} +2 \sqrt {\textit {\_a} a +b}\, b -3 c_{4} a \right )}}d \textit {\_a} = x +c_{6} \] Verified OK.
Maple trace
`Methods for second order ODEs: *** Sublevel 2 *** Methods for second order ODEs: Successful isolation of d^2y/dx^2: 2 solutions were found. Trying to solve each resulting ODE. *** Sublevel 3 *** 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 differential order: 2; missing variables `, `-> Computing symmetries using: way = 3 -> Calling odsolve with the ODE`, (diff(_b(_a), _a))*_b(_a)-(_a*a+b)^(1/2) = 0, _b(_a), HINT = [[4*(_a*a+b)/a, 3*_b]]` symmetry methods on request `, `1st order, trying reduction of order with given symmetries:`[4*(_a*a+b)/a, 3*_b]
✓ Solution by Maple
Time used: 0.046 (sec). Leaf size: 206
dsolve(diff(diff(y(x),x),x)^2-a*y(x)-b=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {b}{a} \\ \sqrt {3}\, a \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {a \left (4 \textit {\_a} \sqrt {a \textit {\_a} +b}\, a +4 \sqrt {a \textit {\_a} +b}\, b -c_{1} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {3}\, a \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {a \left (4 \textit {\_a} \sqrt {a \textit {\_a} +b}\, a +4 \sqrt {a \textit {\_a} +b}\, b -c_{1} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {3}\, a \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {-a \left (4 \textit {\_a} \sqrt {a \textit {\_a} +b}\, a +4 \sqrt {a \textit {\_a} +b}\, b -c_{1} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {3}\, a \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {-a \left (4 \textit {\_a} \sqrt {a \textit {\_a} +b}\, a +4 \sqrt {a \textit {\_a} +b}\, b -c_{1} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.116 (sec). Leaf size: 201
DSolve[-b - a*y[x] + y''[x]^2 == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {(a y(x)+b)^2 \left (1-\frac {4 (a y(x)+b)^{3/2}}{3 a c_1}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\frac {4 (b+a y(x))^{3/2}}{3 a c_1}\right ){}^2}{a^2 \left (-\frac {4 (a y(x)+b)^{3/2}}{3 a}+c_1\right )}&=(x+c_2){}^2,y(x)\right ] \\ \text {Solve}\left [\frac {(a y(x)+b)^2 \left (1+\frac {4 (a y(x)+b)^{3/2}}{3 a c_1}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},-\frac {4 (b+a y(x))^{3/2}}{3 a c_1}\right ){}^2}{a^2 \left (\frac {4 (a y(x)+b)^{3/2}}{3 a}+c_1\right )}&=(x+c_2){}^2,y(x)\right ] \\ \end{align*}