Internal problem ID [9983]
Internal file name [OUTPUT/8930_Monday_June_06_2022_05_59_10_AM_26680355/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1662 (book 6.71).
ODE order: 2.
ODE degree: 1.
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]]
\[ \boxed {8 y^{\prime \prime }+9 {y^{\prime }}^{4}=0} \]
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*} 8 p^{\prime }\left (x \right )+9 p \left (x \right )^{4} = 0 \end {align*}
Which is now solve for \(p(x)\) as first order ode. Integrating both sides gives \begin {align*} \int -\frac {8}{9 p^{4}}d p &= x +c_{1}\\ \frac {8}{27 p^{3}}&=x +c_{1} \end {align*}
Solving for \(p\) gives these solutions \begin {align*} p_1&=\frac {2}{3 \left (x +c_{1} \right )^{\frac {1}{3}}}\\ p_2&=-\frac {1}{3 \left (x +c_{1} \right )^{\frac {1}{3}}}-\frac {i \sqrt {3}}{3 \left (x +c_{1} \right )^{\frac {1}{3}}}\\ p_3&=-\frac {1}{3 \left (x +c_{1} \right )^{\frac {1}{3}}}+\frac {i \sqrt {3}}{3 \left (x +c_{1} \right )^{\frac {1}{3}}} \end {align*}
For solution (1) found earlier, since \(p=y^{\prime }\) then the new first order ode to solve is \begin {align*} y^{\prime } = \frac {2}{3 \left (x +c_{1} \right )^{\frac {1}{3}}} \end {align*}
Integrating both sides gives \begin {align*} y &= \int { \frac {2}{3 \left (x +c_{1} \right )^{\frac {1}{3}}}\,\mathop {\mathrm {d}x}}\\ &= \left (x +c_{1} \right )^{\frac {2}{3}}+c_{2} \end {align*}
Since \(p=y^{\prime }\) then the new first order ode to solve is \begin {align*} y^{\prime } = -\frac {1}{3 \left (x +c_{1} \right )^{\frac {1}{3}}}-\frac {i \sqrt {3}}{3 \left (x +c_{1} \right )^{\frac {1}{3}}} \end {align*}
Integrating both sides gives \begin {align*} y &= \int { -\frac {1+i \sqrt {3}}{3 \left (x +c_{1} \right )^{\frac {1}{3}}}\,\mathop {\mathrm {d}x}}\\ &= \frac {3 \left (-\frac {1}{3}-\frac {i \sqrt {3}}{3}\right ) \left (x +c_{1} \right )^{\frac {2}{3}}}{2}+c_{3} \end {align*}
Since \(p=y^{\prime }\) then the new first order ode to solve is \begin {align*} y^{\prime } = -\frac {1}{3 \left (x +c_{1} \right )^{\frac {1}{3}}}+\frac {i \sqrt {3}}{3 \left (x +c_{1} \right )^{\frac {1}{3}}} \end {align*}
Integrating both sides gives \begin {align*} y &= \int { \frac {i \sqrt {3}-1}{3 \left (x +c_{1} \right )^{\frac {1}{3}}}\,\mathop {\mathrm {d}x}}\\ &= \frac {3 \left (\frac {i \sqrt {3}}{3}-\frac {1}{3}\right ) \left (x +c_{1} \right )^{\frac {2}{3}}}{2}+c_{4} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \left (x +c_{1} \right )^{\frac {2}{3}}+c_{2} \\ \tag{2} y &= \frac {3 \left (-\frac {1}{3}-\frac {i \sqrt {3}}{3}\right ) \left (x +c_{1} \right )^{\frac {2}{3}}}{2}+c_{3} \\ \tag{3} y &= \frac {3 \left (\frac {i \sqrt {3}}{3}-\frac {1}{3}\right ) \left (x +c_{1} \right )^{\frac {2}{3}}}{2}+c_{4} \\ \end{align*}
Verification of solutions
\[ y = \left (x +c_{1} \right )^{\frac {2}{3}}+c_{2} \] Verified OK.
\[ y = \frac {3 \left (-\frac {1}{3}-\frac {i \sqrt {3}}{3}\right ) \left (x +c_{1} \right )^{\frac {2}{3}}}{2}+c_{3} \] Verified OK.
\[ y = \frac {3 \left (\frac {i \sqrt {3}}{3}-\frac {1}{3}\right ) \left (x +c_{1} \right )^{\frac {2}{3}}}{2}+c_{4} \] 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*} 8 p \left (y \right ) \left (\frac {d}{d y}p \left (y \right )\right )+9 p \left (y \right )^{4} = 0 \end {align*}
Which is now solved as first order ode for \(p(y)\). Integrating both sides gives \begin {align*} \int -\frac {8}{9 p^{3}}d p &= y +c_{1}\\ \frac {4}{9 p^{2}}&=y +c_{1} \end {align*}
Solving for \(p\) gives these solutions \begin {align*} p_1&=-\frac {2}{3 \sqrt {y +c_{1}}}\\ p_2&=\frac {2}{3 \sqrt {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*} y^{\prime } = -\frac {2}{3 \sqrt {y+c_{1}}} \end {align*}
Integrating both sides gives \begin{align*} \int -\frac {3 \sqrt {y +c_{1}}}{2}d y &= \int d x \\ -\left (y+c_{1} \right )^{\frac {3}{2}}&=x +c_{2} \\ \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*} y^{\prime } = \frac {2}{3 \sqrt {y+c_{1}}} \end {align*}
Integrating both sides gives \begin{align*} \int \frac {3 \sqrt {y +c_{1}}}{2}d y &= \int d x \\ \left (y+c_{1} \right )^{\frac {3}{2}}&=x +c_{3} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} -\left (y+c_{1} \right )^{\frac {3}{2}} &= x +c_{2} \\ \tag{2} \left (y+c_{1} \right )^{\frac {3}{2}} &= x +c_{3} \\ \end{align*}
Verification of solutions
\[ -\left (y+c_{1} \right )^{\frac {3}{2}} = x +c_{2} \] Verified OK.
\[ \left (y+c_{1} \right )^{\frac {3}{2}} = x +c_{3} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 8 \frac {d}{d x}y^{\prime }+9 {y^{\prime }}^{4}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d x}y^{\prime } \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} u =y^{\prime }\hspace {3pt}\textrm {to reduce order of ODE}\hspace {3pt} \\ {} & {} & 8 u^{\prime }\left (x \right )+9 u \left (x \right )^{4}=0 \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & u^{\prime }\left (x \right )=-\frac {9 u \left (x \right )^{4}}{8} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {u^{\prime }\left (x \right )}{u \left (x \right )^{4}}=-\frac {9}{8} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {u^{\prime }\left (x \right )}{u \left (x \right )^{4}}d x =\int -\frac {9}{8}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{3 u \left (x \right )^{3}}=-\frac {9 x}{8}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} u \left (x \right ) \\ {} & {} & u \left (x \right )=\frac {2 \left (-9 \left (8 c_{1} -9 x \right )^{2}\right )^{\frac {1}{3}}}{3 \left (8 c_{1} -9 x \right )} \\ \bullet & {} & \textrm {Solve 1st ODE for}\hspace {3pt} u \left (x \right ) \\ {} & {} & u \left (x \right )=\frac {2 \left (-9 \left (8 c_{1} -9 x \right )^{2}\right )^{\frac {1}{3}}}{3 \left (8 c_{1} -9 x \right )} \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} u =y^{\prime } \\ {} & {} & y^{\prime }=\frac {2 \left (-9 \left (8 c_{1} -9 x \right )^{2}\right )^{\frac {1}{3}}}{3 \left (8 c_{1} -9 x \right )} \\ \bullet & {} & \textrm {Integrate both sides to solve for}\hspace {3pt} y \\ {} & {} & \int y^{\prime }d x =\int \frac {2 \left (-9 \left (8 c_{1} -9 x \right )^{2}\right )^{\frac {1}{3}}}{3 \left (8 c_{1} -9 x \right )}d x +c_{2} \\ \bullet & {} & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y=-\frac {\left (-9 \left (-8 c_{1} +9 x \right )^{2}\right )^{\frac {1}{3}}}{9}+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 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) = -(9/8)*_b(_a)^4, _b(_a), HINT = [[1, 0], [_a, -(1/3)*_b]]` *** Sublevel 2 *** symmetry methods on request `, `1st order, trying reduction of order with given symmetries:`[1, 0], [_a, -1/3*_b]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 57
dsolve(8*diff(diff(y(x),x),x)+9*diff(y(x),x)^4=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \left (c_{1} +x \right )^{\frac {2}{3}}+c_{2} \\ y \left (x \right ) &= -\frac {i \left (c_{1} +x \right )^{\frac {2}{3}} \sqrt {3}}{2}-\frac {\left (c_{1} +x \right )^{\frac {2}{3}}}{2}+c_{2} \\ y \left (x \right ) &= \frac {i \left (c_{1} +x \right )^{\frac {2}{3}} \sqrt {3}}{2}-\frac {\left (c_{1} +x \right )^{\frac {2}{3}}}{2}+c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.373 (sec). Leaf size: 90
DSolve[9*y'[x]^4 + 8*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2-\frac {1}{3} \sqrt [3]{-\frac {1}{3}} (9 x-8 c_1){}^{2/3} \\ y(x)\to \frac {(9 x-8 c_1){}^{2/3}}{3 \sqrt [3]{3}}+c_2 \\ y(x)\to \frac {1}{9} \left ((-3)^{2/3} (9 x-8 c_1){}^{2/3}+9 c_2\right ) \\ \end{align*}