Internal problem ID [6845]
Internal file name [OUTPUT/6092_Thursday_July_28_2022_04_30_14_AM_58312407/index.tex]
Book: Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam
Publishing Co. NY. 6th edition. 1981.
Section: CHAPTER 16. Nonlinear equations. Section 101. Independent variable missing.
EXERCISES Page 324
Problem number: 27.
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]]
\[ \boxed {x^{2} y^{\prime \prime }+{y^{\prime }}^{2}=0} \]
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^{2} p^{\prime }\left (x \right )+p \left (x \right )^{2} = 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^{2}}{x^{2}} \end {align*}
Where \(f(x)=-\frac {1}{x^{2}}\) and \(g(p)=p^{2}\). Integrating both sides gives \begin{align*} \frac {1}{p^{2}} \,dp &= -\frac {1}{x^{2}} \,d x \\ \int { \frac {1}{p^{2}} \,dp} &= \int {-\frac {1}{x^{2}} \,d x} \\ -\frac {1}{p}&=\frac {1}{x}+c_{1} \\ \end{align*} The solution is \[ -\frac {1}{p \left (x \right )}-\frac {1}{x}-c_{1} = 0 \] 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 {1}{y^{\prime }}-\frac {1}{x}-c_{1} = 0 \end {align*}
Integrating both sides gives \begin {align*} y &= \int { -\frac {x}{c_{1} x +1}\,\mathop {\mathrm {d}x}}\\ &= \frac {-c_{1} x +\ln \left (c_{1} x +1\right )}{c_{1}^{2}}+c_{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {-c_{1} x +\ln \left (c_{1} x +1\right )}{c_{1}^{2}}+c_{2} \\ \end{align*}
Verification of solutions
\[ y = \frac {-c_{1} x +\ln \left (c_{1} x +1\right )}{c_{1}^{2}}+c_{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} y^{\prime \prime }+{y^{\prime }}^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & y^{\prime \prime } \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} u =y^{\prime }\hspace {3pt}\textrm {to reduce order of ODE}\hspace {3pt} \\ {} & {} & x^{2} u^{\prime }\left (x \right )+u \left (x \right )^{2}=0 \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & u^{\prime }\left (x \right )=-\frac {u \left (x \right )^{2}}{x^{2}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {u^{\prime }\left (x \right )}{u \left (x \right )^{2}}=-\frac {1}{x^{2}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {u^{\prime }\left (x \right )}{u \left (x \right )^{2}}d x =\int -\frac {1}{x^{2}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{u \left (x \right )}=\frac {1}{x}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} u \left (x \right ) \\ {} & {} & u \left (x \right )=-\frac {x}{c_{1} x +1} \\ \bullet & {} & \textrm {Solve 1st ODE for}\hspace {3pt} u \left (x \right ) \\ {} & {} & u \left (x \right )=-\frac {x}{c_{1} x +1} \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} u =y^{\prime } \\ {} & {} & y^{\prime }=-\frac {x}{c_{1} x +1} \\ \bullet & {} & \textrm {Integrate both sides to solve for}\hspace {3pt} y \\ {} & {} & \int y^{\prime }d x =\int -\frac {x}{c_{1} x +1}d x +c_{2} \\ \bullet & {} & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y=-\frac {x}{c_{1}}+\frac {\ln \left (c_{1} x +1\right )}{c_{1}^{2}}+c_{2} \end {array} \]
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 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)^2/_a^2, _b(_a), HINT = [[_a, _b]]` *** Sublevel 2 *** symmetry methods on request `, `1st order, trying reduction of order with given symmetries:`[_a, _b]
✓ Solution by Maple
Time used: 0.046 (sec). Leaf size: 21
dsolve(x^2*diff(y(x),x$2)+diff(y(x),x)^2=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {x}{c_{1}}+\frac {\ln \left (c_{1} x -1\right )}{c_{1}^{2}}+c_{2} \]
✓ Solution by Mathematica
Time used: 0.57 (sec). Leaf size: 47
DSolve[x^2*y''[x]+(y'[x])^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x}{c_1}+\frac {\log (1+c_1 x)}{c_1{}^2}+c_2 \\ y(x)\to c_2 \\ y(x)\to -\frac {x^2}{2}+c_2 \\ \end{align*}