Internal problem ID [6843]
Internal file name [OUTPUT/6090_Thursday_July_28_2022_04_30_10_AM_70078846/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: 25.
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]]
\[ \boxed {y^{\prime \prime }-{\mathrm e}^{x} {y^{\prime }}^{2}=0} \]
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*} p^{\prime }\left (x \right )-{\mathrm e}^{x} 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)\\ &= {\mathrm e}^{x} p^{2} \end {align*}
Where \(f(x)={\mathrm e}^{x}\) and \(g(p)=p^{2}\). Integrating both sides gives \begin{align*} \frac {1}{p^{2}} \,dp &= {\mathrm e}^{x} \,d x \\ \int { \frac {1}{p^{2}} \,dp} &= \int {{\mathrm e}^{x} \,d x} \\ -\frac {1}{p}&={\mathrm e}^{x}+c_{1} \\ \end{align*} The solution is \[ -\frac {1}{p \left (x \right )}-{\mathrm e}^{x}-c_{1} = 0 \] 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 }}-{\mathrm e}^{x}-c_{1} = 0 \end {align*}
Integrating both sides gives \begin {align*} y &= \int { -\frac {1}{{\mathrm e}^{x}+c_{1}}\,\mathop {\mathrm {d}x}}\\ &= \frac {-x +\ln \left ({\mathrm e}^{x}+c_{1} \right )}{c_{1}}+c_{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {-x +\ln \left ({\mathrm e}^{x}+c_{1} \right )}{c_{1}}+c_{2} \\ \end{align*}
Verification of solutions
\[ y = \frac {-x +\ln \left ({\mathrm e}^{x}+c_{1} \right )}{c_{1}}+c_{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime }-{\mathrm e}^{x} {y^{\prime }}^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & y^{\prime \prime } \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} u =y^{\prime }\hspace {3pt}\textrm {to reduce order of ODE}\hspace {3pt} \\ {} & {} & u^{\prime }\left (x \right )-{\mathrm e}^{x} u \left (x \right )^{2}=0 \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & u^{\prime }\left (x \right )={\mathrm e}^{x} u \left (x \right )^{2} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {u^{\prime }\left (x \right )}{u \left (x \right )^{2}}={\mathrm e}^{x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {u^{\prime }\left (x \right )}{u \left (x \right )^{2}}d x =\int {\mathrm e}^{x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{u \left (x \right )}={\mathrm e}^{x}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} u \left (x \right ) \\ {} & {} & u \left (x \right )=-\frac {1}{{\mathrm e}^{x}+c_{1}} \\ \bullet & {} & \textrm {Solve 1st ODE for}\hspace {3pt} u \left (x \right ) \\ {} & {} & u \left (x \right )=-\frac {1}{{\mathrm e}^{x}+c_{1}} \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} u =y^{\prime } \\ {} & {} & y^{\prime }=-\frac {1}{{\mathrm e}^{x}+c_{1}} \\ \bullet & {} & \textrm {Integrate both sides to solve for}\hspace {3pt} y \\ {} & {} & \int y^{\prime }d x =\int -\frac {1}{{\mathrm e}^{x}+c_{1}}d x +c_{2} \\ \bullet & {} & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y=\frac {\ln \left ({\mathrm e}^{x}+c_{1} \right )}{c_{1}}-\frac {\ln \left ({\mathrm e}^{x}\right )}{c_{1}}+c_{2} \end {array} \]
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) = exp(_a)*_b(_a)^2, _b(_a), HINT = [[1, -_b]]` *** Sublevel 2 *** symmetry methods on request `, `1st order, trying reduction of order with given symmetries:`[1, -_b]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 24
dsolve(diff(y(x),x$2)=exp(x)*diff(y(x),x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{2} c_{1} -\ln \left ({\mathrm e}^{x}-c_{1} \right )+\ln \left ({\mathrm e}^{x}\right )}{c_{1}} \]
✓ Solution by Mathematica
Time used: 0.985 (sec). Leaf size: 37
DSolve[y''[x]==Exp[x](y'[x])^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-x+\log \left (e^x+c_1\right )+c_1 c_2}{c_1} \\ y(x)\to \text {Indeterminate} \\ y(x)\to c_2 \\ \end{align*}