Internal problem ID [4845]
Internal file name [OUTPUT/4338_Sunday_June_05_2022_01_02_58_PM_31761600/index.tex]
Book: Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley.
2006
Section: Chapter 8, Ordinary differential equations. Section 7. Other second-Order equations.
page 435
Problem number: 4.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_ode_missing_y"
Maple gives the following as the ode type
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]
\[ \boxed {x y^{\prime \prime }-y^{\prime }-{y^{\prime }}^{3}=0} \]
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*} x p^{\prime }\left (x \right )+\left (-1-p \left (x \right )^{2}\right ) p \left (x \right ) = 0 \end {align*}
Which is now solve for \(p(x)\) as first order ode. In canonical form the ODE is \begin {align*} p' &= F(x,p)\\ &= f( x) g(p)\\ &= \frac {p \left (p^{2}+1\right )}{x} \end {align*}
Where \(f(x)=\frac {1}{x}\) and \(g(p)=p \left (p^{2}+1\right )\). Integrating both sides gives \begin{align*} \frac {1}{p \left (p^{2}+1\right )} \,dp &= \frac {1}{x} \,d x \\ \int { \frac {1}{p \left (p^{2}+1\right )} \,dp} &= \int {\frac {1}{x} \,d x} \\ \ln \left (p \right )-\frac {\ln \left (p^{2}+1\right )}{2}&=\ln \left (x \right )+c_{1} \\ \end{align*} Raising both side to exponential gives \begin {align*} {\mathrm e}^{\ln \left (p \right )-\frac {\ln \left (p^{2}+1\right )}{2}} &= {\mathrm e}^{\ln \left (x \right )+c_{1}} \end {align*}
Which simplifies to \begin {align*} \frac {p}{\sqrt {p^{2}+1}} &= c_{2} x \end {align*}
Since \(p=y^{\prime }\) then the new first order ode to solve is \begin {align*} y^{\prime } = c_{2} x \sqrt {-\frac {1}{c_{2}^{2} x^{2}-1}} \end {align*}
Integrating both sides gives \begin {align*} y &= \int { c_{2} x \sqrt {-\frac {1}{c_{2}^{2} x^{2}-1}}\,\mathop {\mathrm {d}x}}\\ &= \frac {\sqrt {-\frac {1}{c_{2}^{2} x^{2}-1}}\, \left (c_{2}^{2} x^{2}-1\right )}{c_{2}}+c_{3} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\sqrt {-\frac {1}{c_{2}^{2} x^{2}-1}}\, \left (c_{2}^{2} x^{2}-1\right )}{c_{2}}+c_{3} \\ \end{align*}
Verification of solutions
\[ y = \frac {\sqrt {-\frac {1}{c_{2}^{2} x^{2}-1}}\, \left (c_{2}^{2} x^{2}-1\right )}{c_{2}}+c_{3} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y^{\prime \prime }+\left (-1-{y^{\prime }}^{2}\right ) y^{\prime }=0 \\ \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} \\ {} & {} & x u^{\prime }\left (x \right )+\left (-1-u \left (x \right )^{2}\right ) u \left (x \right )=0 \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & u^{\prime }\left (x \right )=-\frac {\left (-1-u \left (x \right )^{2}\right ) u \left (x \right )}{x} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {u^{\prime }\left (x \right )}{\left (-1-u \left (x \right )^{2}\right ) u \left (x \right )}=-\frac {1}{x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {u^{\prime }\left (x \right )}{\left (-1-u \left (x \right )^{2}\right ) u \left (x \right )}d x =\int -\frac {1}{x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\ln \left (u \left (x \right )^{2}+1\right )}{2}-\ln \left (u \left (x \right )\right )=-\ln \left (x \right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} u \left (x \right ) \\ {} & {} & \left \{u \left (x \right )=\frac {x}{\sqrt {\left ({\mathrm e}^{c_{1}}\right )^{2}-x^{2}}}, u \left (x \right )=-\frac {x}{\sqrt {\left ({\mathrm e}^{c_{1}}\right )^{2}-x^{2}}}\right \} \\ \bullet & {} & \textrm {Solve 1st ODE for}\hspace {3pt} u \left (x \right ) \\ {} & {} & u \left (x \right )=\frac {x}{\sqrt {\left ({\mathrm e}^{c_{1}}\right )^{2}-x^{2}}} \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} u =y^{\prime } \\ {} & {} & y^{\prime }=\frac {x}{\sqrt {\left ({\mathrm e}^{c_{1}}\right )^{2}-x^{2}}} \\ \bullet & {} & \textrm {Integrate both sides to solve for}\hspace {3pt} y \\ {} & {} & \int y^{\prime }d x =\int \frac {x}{\sqrt {\left ({\mathrm e}^{c_{1}}\right )^{2}-x^{2}}}d x +c_{2} \\ \bullet & {} & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y=-\sqrt {\left ({\mathrm e}^{c_{1}}\right )^{2}-x^{2}}+c_{2} \\ \bullet & {} & \textrm {Solve 2nd ODE for}\hspace {3pt} u \left (x \right ) \\ {} & {} & u \left (x \right )=-\frac {x}{\sqrt {\left ({\mathrm e}^{c_{1}}\right )^{2}-x^{2}}} \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} u =y^{\prime } \\ {} & {} & y^{\prime }=-\frac {x}{\sqrt {\left ({\mathrm e}^{c_{1}}\right )^{2}-x^{2}}} \\ \bullet & {} & \textrm {Integrate both sides to solve for}\hspace {3pt} y \\ {} & {} & \int y^{\prime }d x =\int -\frac {x}{\sqrt {\left ({\mathrm e}^{c_{1}}\right )^{2}-x^{2}}}d x +c_{2} \\ \bullet & {} & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y=\sqrt {\left ({\mathrm e}^{c_{1}}\right )^{2}-x^{2}}+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 a change of variables {x -> y(x), y(x) -> x} 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)*(_b(_a)^2+1)/_a, _b(_a), HINT = [[_a, 0]]` *** Sublevel 2 *** symmetry methods on request `, `1st order, trying reduction of order with given symmetries:`[_a, 0]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 31
dsolve(x*diff(y(x),x$2)=diff(y(x),x)+(diff(y(x),x))^3,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\sqrt {-x^{2}+c_{1}}+c_{2} \\ y \left (x \right ) &= \sqrt {-x^{2}+c_{1}}+c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.486 (sec). Leaf size: 103
DSolve[x*y''[x]==y'[x]+(y'[x])^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2-i e^{-c_1} \sqrt {-1+e^{2 c_1} x^2} \\ y(x)\to i e^{-c_1} \sqrt {-1+e^{2 c_1} x^2}+c_2 \\ y(x)\to c_2-i \sqrt {x^2} \\ y(x)\to i \sqrt {x^2}+c_2 \\ \end{align*}