Internal problem ID [15202]
Internal file name [OUTPUT/15203_Tuesday_April_23_2024_04_54_01_PM_24959990/index.tex]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 2 (Higher order ODE’s). Section 14. Differential equations admitting of
depression of their order. Exercises page 107
Problem number: 345.
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]]
\[ \boxed {y^{\prime \prime }-\sqrt {y^{\prime }+1}=0} \]
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 )-\sqrt {p \left (x \right )+1} = 0 \end {align*}
Which is now solve for \(p(x)\) as first order ode. Integrating both sides gives \begin{align*} \int \frac {1}{\sqrt {p +1}}d p &= \int d x \\ 2 \sqrt {p \left (x \right )+1}&=x +c_{1} \\ \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_{2} \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_{2} \\ \end{align*}
Verification of solutions
\[ y = \frac {x \left (3 c_{1}^{2}+3 c_{1} x +x^{2}-12\right )}{12}+c_{2} \] 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 ) \left (\frac {d}{d y}p \left (y \right )\right ) = \sqrt {p \left (y \right )+1} \end {align*}
Which is now solved as first order ode for \(p(y)\). Integrating both sides gives \begin{align*} \int \frac {p}{\sqrt {p +1}}d p &= \int d y \\ \frac {2 \sqrt {p \left (y \right )+1}\, \left (p \left (y \right )-2\right )}{3}&=y +c_{1} \\ \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 \sqrt {y^{\prime }+1}\, \left (y^{\prime }-2\right )}{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_{2} \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}+\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}}+16-8 \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} = c_{3} +x \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}-\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}}-16+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*}
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_{2} \\ \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}+\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}}+16-8 \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} &= c_{3} +x \\ \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}-\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}}-16+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*}
Verification of solutions
\[ \int _{}^{y}\frac {4 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}}{\left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}}+4 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}+16}d \textit {\_a} = x +c_{2} \] Verified OK.
\[ \int _{}^{y}-\frac {8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}}{i \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}} \sqrt {3}+\left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}}+16-8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}-16 i \sqrt {3}}d \textit {\_a} = c_{3} +x \] Verified OK.
\[ \int _{}^{y}\frac {8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}}{i \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}} \sqrt {3}-\left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}}-16+8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}-16 i \sqrt {3}}d \textit {\_a} = x +c_{4} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime }=\sqrt {y^{\prime }+1} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & y^{\prime \prime } \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} u =y^{\prime }\hspace {3pt}\textrm {to reduce order of ODE}\hspace {3pt} \\ {} & {} & u^{\prime }\left (x \right )=\sqrt {u \left (x \right )+1} \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & u^{\prime }\left (x \right )=\sqrt {u \left (x \right )+1} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {u^{\prime }\left (x \right )}{\sqrt {u \left (x \right )+1}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {u^{\prime }\left (x \right )}{\sqrt {u \left (x \right )+1}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & 2 \sqrt {u \left (x \right )+1}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} u \left (x \right ) \\ {} & {} & u \left (x \right )=\frac {1}{4} c_{1}^{2}+\frac {1}{2} c_{1} x +\frac {1}{4} x^{2}-1 \\ \bullet & {} & \textrm {Solve 1st ODE for}\hspace {3pt} u \left (x \right ) \\ {} & {} & u \left (x \right )=\frac {1}{4} c_{1}^{2}+\frac {1}{2} c_{1} x +\frac {1}{4} x^{2}-1 \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} u =y^{\prime } \\ {} & {} & y^{\prime }=\frac {1}{4} c_{1}^{2}+\frac {1}{2} c_{1} x +\frac {1}{4} x^{2}-1 \\ \bullet & {} & \textrm {Integrate both sides to solve for}\hspace {3pt} y \\ {} & {} & \int y^{\prime }d x =\int \left (\frac {1}{4} c_{1}^{2}+\frac {1}{2} c_{1} x +\frac {1}{4} x^{2}-1\right )d x +c_{2} \\ \bullet & {} & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y=\frac {x^{3}}{12}+\frac {c_{1} x^{2}}{4}+\frac {x \left (c_{1} +2\right ) \left (c_{1} -2\right )}{4}+c_{2} \end {array} \]
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 2nd order ODE linearizable_by_differentiation -> Calling odsolve with the ODE`, diff(diff(diff(y(x), x), x), x)-1/2, y(x)` *** Sublevel 2 *** Methods for third order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful 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)+1)^(1/2), _b(_a), HINT = [[1, 0], [_a, 2+2*_b]]` *** Sublevel 2 *** symmetry methods on request `, `1st order, trying reduction of order with given symmetries:`[1, 0], [_a, 2+2*_b]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 34
dsolve(diff(y(x),x$2)=sqrt(1+diff(y(x),x)),y(x), singsol=all)
\begin{align*} y &= -x +c_{1} \\ y &= \frac {1}{12} x^{3}+\frac {1}{4} c_{1} x^{2}+\frac {1}{4} c_{1}^{2} x -x +c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.052 (sec). Leaf size: 30
DSolve[y''[x]==Sqrt[1+y'[x]],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{12} x \left (x^2+3 c_1 x+3 \left (-4+c_1{}^2\right )\right )+c_2 \]