Internal problem ID [7405]
Internal file name [OUTPUT/6372_Sunday_June_05_2022_04_42_02_PM_299138/index.tex]
Book: Second order enumerated odes
Section: section 1
Problem number: 16.
ODE order: 2.
ODE degree: 2.
The type(s) of ODE detected by this program : "second_order_ode_missing_x", "second_order_ode_missing_y"
Maple gives the following as the ode type
[[_2nd_order, _missing_x]]
\[ \boxed {{y^{\prime \prime }}^{2}+y^{\prime }=1} \]
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 )}^{2}+p \left (x \right )-1 = 0 \end {align*}
Which is now solve for \(p(x)\) as first order ode. Solving the given ode for \(p^{\prime }\left (x \right )\) results in \(2\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} p^{\prime }\left (x \right )&=\sqrt {1-p \left (x \right )} \tag {1} \\ p^{\prime }\left (x \right )&=-\sqrt {1-p \left (x \right )} \tag {2} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Integrating both sides gives \begin{align*} \int \frac {1}{\sqrt {1-p}}d p &= \int d x \\ -2 \sqrt {1-p \left (x \right )}&=x +c_{1} \\ \end{align*} Solving equation (2)
Integrating both sides gives \begin{align*} \int -\frac {1}{\sqrt {1-p}}d p &= \int d x \\ 2 \sqrt {1-p \left (x \right )}&=x +c_{2} \\ \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*} -2 \sqrt {-y^{\prime }+1} = x +c_{1} \end {align*}
Integrating both sides gives \begin {align*} y &= \int { -\frac {1}{4} c_{1}^{2}-\frac {1}{2} c_{1} x -\frac {1}{4} x^{2}+1\,\mathop {\mathrm {d}x}}\\ &= -\frac {x \left (3 c_{1}^{2}+3 c_{1} x +x^{2}-12\right )}{12}+c_{3} \end {align*}
For solution (2) found earlier, since \(p=y^{\prime }\) then we now have a new first order ode to solve which is \begin {align*} 2 \sqrt {-y^{\prime }+1} = x +c_{2} \end {align*}
Integrating both sides gives \begin {align*} y &= \int { -\frac {1}{4} c_{2}^{2}-\frac {1}{2} x c_{2} -\frac {1}{4} x^{2}+1\,\mathop {\mathrm {d}x}}\\ &= -\frac {x \left (3 c_{2}^{2}+3 x c_{2} +x^{2}-12\right )}{12}+c_{4} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {x \left (3 c_{1}^{2}+3 c_{1} x +x^{2}-12\right )}{12}+c_{3} \\ \tag{2} y &= -\frac {x \left (3 c_{2}^{2}+3 x c_{2} +x^{2}-12\right )}{12}+c_{4} \\ \end{align*}
Verification of solutions
\[ y = -\frac {x \left (3 c_{1}^{2}+3 c_{1} x +x^{2}-12\right )}{12}+c_{3} \] Verified OK.
\[ y = -\frac {x \left (3 c_{2}^{2}+3 x c_{2} +x^{2}-12\right )}{12}+c_{4} \] Verified OK.
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*} p \left (y \right )^{2} \left (\frac {d}{d y}p \left (y \right )\right )^{2}+p \left (y \right ) = 1 \end {align*}
Which is now solved as first order ode for \(p(y)\). Solving the given ode for \(\frac {d}{d y}p \left (y \right )\) results in \(2\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} \frac {d}{d y}p \left (y \right )&=\frac {\sqrt {1-p \left (y \right )}}{p \left (y \right )} \tag {1} \\ \frac {d}{d y}p \left (y \right )&=-\frac {\sqrt {1-p \left (y \right )}}{p \left (y \right )} \tag {2} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Integrating both sides gives \begin{align*} \int \frac {p}{\sqrt {1-p}}d p &= \int d y \\ -\frac {2 \left (p \left (y \right )+2\right ) \sqrt {1-p \left (y \right )}}{3}&=y +c_{1} \\ \end{align*} Solving equation (2)
Integrating both sides gives \begin{align*} \int -\frac {p}{\sqrt {1-p}}d p &= \int d y \\ \frac {2 \left (p \left (y \right )+2\right ) \sqrt {1-p \left (y \right )}}{3}&=y +c_{2} \\ \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 {2 \left (y^{\prime }+2\right ) \sqrt {-y^{\prime }+1}}{3} = y+c_{1} \end {align*}
Solving the given ode for \(y^{\prime }\) results in \(3\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} y^{\prime }&=-{\left (\frac {\left (6 y+6 c_{1} +2 \sqrt {-16+9 y^{2}+18 y c_{1} +9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (6 y+6 c_{1} +2 \sqrt {-16+9 y^{2}+18 y c_{1} +9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}^{2}+1 \tag {1} \\ y^{\prime }&=-{\left (-\frac {\left (6 y+6 c_{1} +2 \sqrt {-16+9 y^{2}+18 y c_{1} +9 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {1}{\left (6 y+6 c_{1} +2 \sqrt {-16+9 y^{2}+18 y c_{1} +9 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (6 y+6 c_{1} +2 \sqrt {-16+9 y^{2}+18 y c_{1} +9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (6 y+6 c_{1} +2 \sqrt {-16+9 y^{2}+18 y c_{1} +9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right )}^{2}+1 \tag {2} \\ y^{\prime }&=-{\left (-\frac {\left (6 y+6 c_{1} +2 \sqrt {-16+9 y^{2}+18 y c_{1} +9 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {1}{\left (6 y+6 c_{1} +2 \sqrt {-16+9 y^{2}+18 y c_{1} +9 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (6 y+6 c_{1} +2 \sqrt {-16+9 y^{2}+18 y c_{1} +9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (6 y+6 c_{1} +2 \sqrt {-16+9 y^{2}+18 y c_{1} +9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right )}^{2}+1 \tag {3} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Integrating both sides gives \begin {align*} \int _{}^{y}-\frac {4 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}}{\left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}}+4 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}+16}d \textit {\_a} = x +c_{3} \end {align*}
Solving equation (2)
Integrating both sides gives \begin {align*} \int _{}^{y}\frac {8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}}{i \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}} \sqrt {3}+16+\left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}}-8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}-16 i \sqrt {3}}d \textit {\_a} = x +c_{4} \end {align*}
Solving equation (3)
Integrating both sides gives \begin {align*} \int _{}^{y}-\frac {8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}}{i \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}} \sqrt {3}-16-\left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}}+8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}-16 i \sqrt {3}}d \textit {\_a} = x +c_{5} \end {align*}
For solution (2) found earlier, since \(p=y^{\prime }\) then we now have a new first order ode to solve which is \begin {align*} \frac {2 \left (y^{\prime }+2\right ) \sqrt {-y^{\prime }+1}}{3} = y+c_{2} \end {align*}
Solving the given ode for \(y^{\prime }\) results in \(3\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} y^{\prime }&=-{\left (\frac {\left (-6 y-6 c_{2} +2 \sqrt {-16+9 y^{2}+18 y c_{2} +9 c_{2}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (-6 y-6 c_{2} +2 \sqrt {-16+9 y^{2}+18 y c_{2} +9 c_{2}^{2}}\right )^{\frac {1}{3}}}\right )}^{2}+1 \tag {1} \\ y^{\prime }&=-{\left (-\frac {\left (-6 y-6 c_{2} +2 \sqrt {-16+9 y^{2}+18 y c_{2} +9 c_{2}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {1}{\left (-6 y-6 c_{2} +2 \sqrt {-16+9 y^{2}+18 y c_{2} +9 c_{2}^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-6 y-6 c_{2} +2 \sqrt {-16+9 y^{2}+18 y c_{2} +9 c_{2}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-6 y-6 c_{2} +2 \sqrt {-16+9 y^{2}+18 y c_{2} +9 c_{2}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right )}^{2}+1 \tag {2} \\ y^{\prime }&=-{\left (-\frac {\left (-6 y-6 c_{2} +2 \sqrt {-16+9 y^{2}+18 y c_{2} +9 c_{2}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {1}{\left (-6 y-6 c_{2} +2 \sqrt {-16+9 y^{2}+18 y c_{2} +9 c_{2}^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-6 y-6 c_{2} +2 \sqrt {-16+9 y^{2}+18 y c_{2} +9 c_{2}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-6 y-6 c_{2} +2 \sqrt {-16+9 y^{2}+18 y c_{2} +9 c_{2}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right )}^{2}+1 \tag {3} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Integrating both sides gives \begin {align*} \int _{}^{y}-\frac {4 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {2}{3}}}{\left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {4}{3}}+4 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {2}{3}}+16}d \textit {\_a} = x +c_{6} \end {align*}
Solving equation (2)
Integrating both sides gives \begin {align*} \int _{}^{y}\frac {8 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {2}{3}}}{i \sqrt {3}\, \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {4}{3}}+\left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {4}{3}}-16 i \sqrt {3}-8 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {2}{3}}+16}d \textit {\_a} = x +c_{7} \end {align*}
Solving equation (3)
Integrating both sides gives \begin {align*} \int _{}^{y}-\frac {8 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {2}{3}}}{i \sqrt {3}\, \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {4}{3}}-\left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {4}{3}}-16 i \sqrt {3}+8 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {2}{3}}-16}d \textit {\_a} = x +c_{8} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}-\frac {4 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}}{\left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}}+4 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}+16}d \textit {\_a} &= x +c_{3} \\ \tag{2} \int _{}^{y}\frac {8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}}{i \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}} \sqrt {3}+16+\left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}}-8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}-16 i \sqrt {3}}d \textit {\_a} &= x +c_{4} \\ \tag{3} \int _{}^{y}-\frac {8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}}{i \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}} \sqrt {3}-16-\left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}}+8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}-16 i \sqrt {3}}d \textit {\_a} &= x +c_{5} \\ \tag{4} \int _{}^{y}-\frac {4 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {2}{3}}}{\left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {4}{3}}+4 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {2}{3}}+16}d \textit {\_a} &= x +c_{6} \\ \tag{5} \int _{}^{y}\frac {8 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {2}{3}}}{i \sqrt {3}\, \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {4}{3}}+\left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {4}{3}}-16 i \sqrt {3}-8 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {2}{3}}+16}d \textit {\_a} &= x +c_{7} \\ \tag{6} \int _{}^{y}-\frac {8 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {2}{3}}}{i \sqrt {3}\, \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {4}{3}}-\left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {4}{3}}-16 i \sqrt {3}+8 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {2}{3}}-16}d \textit {\_a} &= x +c_{8} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}-\frac {4 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}}{\left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}}+4 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}+16}d \textit {\_a} = x +c_{3} \] Verified OK.
\[ \int _{}^{y}\frac {8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}}{i \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}} \sqrt {3}+16+\left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}}-8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}-16 i \sqrt {3}}d \textit {\_a} = x +c_{4} \] Verified OK.
\[ \int _{}^{y}-\frac {8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}}{i \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}} \sqrt {3}-16-\left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}}+8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}-16 i \sqrt {3}}d \textit {\_a} = x +c_{5} \] Verified OK.
\[ \int _{}^{y}-\frac {4 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {2}{3}}}{\left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {4}{3}}+4 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {2}{3}}+16}d \textit {\_a} = x +c_{6} \] Verified OK.
\[ \int _{}^{y}\frac {8 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {2}{3}}}{i \sqrt {3}\, \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {4}{3}}+\left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {4}{3}}-16 i \sqrt {3}-8 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {2}{3}}+16}d \textit {\_a} = x +c_{7} \] Verified OK.
\[ \int _{}^{y}-\frac {8 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {2}{3}}}{i \sqrt {3}\, \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {4}{3}}-\left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {4}{3}}-16 i \sqrt {3}+8 \left (-6 \textit {\_a} -6 c_{2} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{2} +9 c_{2}^{2}-16}\right )^{\frac {2}{3}}-16}d \textit {\_a} = x +c_{8} \] 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 <- 2nd order ODE linearizable_by_differentiation successful ------------------- * Tackling next 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 <- 2nd order ODE linearizable_by_differentiation successful -> Calling odsolve with the ODE`, diff(y(x), x) = 1, y(x), singsol = none` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful`
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 30
dsolve(diff(y(x),x$2)^2+diff(y(x),x)=1,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= x +c_{1} \\ y \left (x \right ) &= -\frac {1}{12} x^{3}+\frac {1}{2} c_{1} x^{2}-x \,c_{1}^{2}+x +c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.027 (sec). Leaf size: 67
DSolve[(y''[x])^2+y'[x]==1,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x^3}{12}-\frac {c_1 x^2}{4}+x-\frac {c_1{}^2 x}{4}+c_2 \\ y(x)\to -\frac {x^3}{12}+\frac {c_1 x^2}{4}+x-\frac {c_1{}^2 x}{4}+c_2 \\ \end{align*}