Internal problem ID [13516]
Internal file name [OUTPUT/12688_Friday_February_16_2024_12_10_59_AM_29537943/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 13. Higher order equations: Extending first order concepts. Additional exercises
page 259
Problem number: 13.7 (e).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_integrable_as_is", "second_order_ode_missing_x", "exact nonlinear second order ode"
Maple gives the following as the ode type
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]]
\[ \boxed {y^{\prime \prime }+y^{\prime } {\mathrm e}^{-y}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 2] \end {align*}
Integrating both sides of the ODE w.r.t \(x\) gives \begin {align*} \int \left (y^{\prime \prime }+y^{\prime } {\mathrm e}^{-y}\right )d x &= 0 \\ -{\mathrm e}^{-y}+y^{\prime } = c_{1} \end {align*}
Which is now solved for \(y\). Integrating both sides gives \begin {align*} \int \frac {{\mathrm e}^{y}}{{\mathrm e}^{y} c_{1} +1}d y &= x +c_{2}\\ \frac {\ln \left ({\mathrm e}^{y} c_{1} +1\right )}{c_{1}}&=x +c_{2} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\ln \left (\frac {{\mathrm e}^{c_{1} c_{2} +c_{1} x}-1}{c_{1}}\right )\\ &=\ln \left (\frac {c_{2} {\mathrm e}^{c_{1} x}-1}{c_{1}}\right ) \end {align*}
Initial conditions are used to solve for the constants of integration.
Looking at the above solution \begin {align*} y = \ln \left (\frac {c_{2} {\mathrm e}^{c_{1} x}-1}{c_{1}}\right ) \tag {1} \end {align*}
Initial conditions are now substituted in the above solution. This will generate the required equations to solve for the integration constants. substituting \(y = 0\) and \(x = 0\) in the above gives \begin {align*} 0 = \ln \left (\frac {c_{2} -1}{c_{1}}\right )\tag {1A} \end {align*}
Taking derivative of the solution gives \begin {align*} y^{\prime } = \frac {c_{2} {\mathrm e}^{c_{1} x} c_{1}}{c_{2} {\mathrm e}^{c_{1} x}-1} \end {align*}
substituting \(y^{\prime } = 2\) and \(x = 0\) in the above gives \begin {align*} 2 = \frac {c_{2} c_{1}}{c_{2} -1}\tag {2A} \end {align*}
Equations {1A,2A} are now solved for \(\{c_{1}, c_{2}\}\). Solving for the constants gives \begin {align*} c_{1}&=1\\ c_{2}&=2 \end {align*}
Substituting these values back in above solution results in \begin {align*} y = \ln \left (2 \,{\mathrm e}^{x}-1\right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \ln \left (2 \,{\mathrm e}^{x}-1\right ) \\ \end{align*}
Verification of solutions
\[ y = \ln \left (2 \,{\mathrm e}^{x}-1\right ) \] 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 )+p \left (y \right ) {\mathrm e}^{-y} = 0 \end {align*}
Which is now solved as first order ode for \(p(y)\). Integrating both sides gives \begin {align*} p \left (y \right ) &= \int { -{\mathrm e}^{-y}\,\mathop {\mathrm {d}y}}\\ &= {\mathrm e}^{-y}+c_{1} \end {align*}
Initial conditions are used to solve for \(c_{1}\). Substituting \(y=0\) and \(p=2\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} 2 = 1+c_{1} \end {align*}
The solutions are \begin {align*} c_{1} = 1 \end {align*}
Trying the constant \begin {align*} c_{1} = 1 \end {align*}
Substituting this in the general solution gives \begin {align*} p \left (y \right )&={\mathrm e}^{-y}+1 \end {align*}
The constant \(c_{1} = 1\) gives valid solution.
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 } = {\mathrm e}^{-y}+1 \end {align*}
Integrating both sides gives \begin {align*} \int \frac {1}{{\mathrm e}^{-y}+1}d y &= \int {dx}\\ y +\ln \left ({\mathrm e}^{-y}+1\right )&= x +c_{2} \end {align*}
Initial conditions are used to solve for \(c_{2}\). Substituting \(x=0\) and \(y=0\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} \ln \left (2\right ) = c_{2} \end {align*}
The solutions are \begin {align*} c_{2} = \ln \left (2\right ) \end {align*}
Trying the constant \begin {align*} c_{2} = \ln \left (2\right ) \end {align*}
Substituting \(c_{2}\) found above in the general solution gives \begin {align*} y +\ln \left ({\mathrm e}^{-y}+1\right ) = x +\ln \left (2\right ) \end {align*}
The constant \(c_{2} = \ln \left (2\right )\) gives valid solution.
Initial conditions are used to solve for the constants of integration.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \ln \left (2 \,{\mathrm e}^{x}-1\right ) \\ \end{align*}
Verification of solutions
\[ y = \ln \left (2 \,{\mathrm e}^{x}-1\right ) \] Verified OK.
Writing the ode as \[ y^{\prime \prime }+y^{\prime } {\mathrm e}^{-y} = 0 \] Integrating both sides of the ODE w.r.t \(x\) gives \begin {align*} \int \left (y^{\prime \prime }+y^{\prime } {\mathrm e}^{-y}\right )d x &= 0 \\ -{\mathrm e}^{-y}+y^{\prime } = c_{1} \end {align*}
Which is now solved for \(y\). Integrating both sides gives \begin {align*} \int \frac {{\mathrm e}^{y}}{{\mathrm e}^{y} c_{1} +1}d y &= x +c_{2}\\ \frac {\ln \left ({\mathrm e}^{y} c_{1} +1\right )}{c_{1}}&=x +c_{2} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\ln \left (\frac {{\mathrm e}^{c_{1} c_{2} +c_{1} x}-1}{c_{1}}\right )\\ &=\ln \left (\frac {c_{2} {\mathrm e}^{c_{1} x}-1}{c_{1}}\right ) \end {align*}
Initial conditions are used to solve for the constants of integration.
Looking at the above solution \begin {align*} y = \ln \left (\frac {c_{2} {\mathrm e}^{c_{1} x}-1}{c_{1}}\right ) \tag {1} \end {align*}
Initial conditions are now substituted in the above solution. This will generate the required equations to solve for the integration constants. substituting \(y = 0\) and \(x = 0\) in the above gives \begin {align*} 0 = \ln \left (\frac {c_{2} -1}{c_{1}}\right )\tag {1A} \end {align*}
Taking derivative of the solution gives \begin {align*} y^{\prime } = \frac {c_{2} {\mathrm e}^{c_{1} x} c_{1}}{c_{2} {\mathrm e}^{c_{1} x}-1} \end {align*}
substituting \(y^{\prime } = 2\) and \(x = 0\) in the above gives \begin {align*} 2 = \frac {c_{2} c_{1}}{c_{2} -1}\tag {2A} \end {align*}
Equations {1A,2A} are now solved for \(\{c_{1}, c_{2}\}\). Solving for the constants gives \begin {align*} c_{1}&=1\\ c_{2}&=2 \end {align*}
Substituting these values back in above solution results in \begin {align*} y = \ln \left (2 \,{\mathrm e}^{x}-1\right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \ln \left (2 \,{\mathrm e}^{x}-1\right ) \\ \end{align*}
Verification of solutions
\[ y = \ln \left (2 \,{\mathrm e}^{x}-1\right ) \] Verified OK.
An exact non-linear second order ode has the form \begin {align*} a_{2} \left (x , y, y^{\prime }\right ) y^{\prime \prime }+a_{1} \left (x , y, y^{\prime }\right ) y^{\prime }+a_{0} \left (x , y, y^{\prime }\right )&=0 \end {align*}
Where the following conditions are satisfied \begin {align*} \frac {\partial a_2}{\partial y} &= \frac {\partial a_1}{\partial y'}\\ \frac {\partial a_2}{\partial x} &= \frac {\partial a_0}{\partial y'}\\ \frac {\partial a_1}{\partial x} &= \frac {\partial a_0}{\partial y} \end {align*}
Looking at the the ode given we see that \begin {align*} a_2 &= 1\\ a_1 &= {\mathrm e}^{-y}\\ a_0 &= 0 \end {align*}
Applying the conditions to the above shows this is a nonlinear exact second order ode. Therefore it can be reduced to first order ode given by \begin {align*} \int {a_2\,d y'} + \int {a_1\,d y} + \int {a_0\,d x} &= c_{1}\\ \int {1\,d y'} + \int {{\mathrm e}^{-y}\,d y} + \int {0\,d x} &= c_{1} \end {align*}
Which results in \begin {align*} -{\mathrm e}^{-y}+y^{\prime } = c_{1} \end {align*}
Which is now solved Integrating both sides gives \begin {align*} \int \frac {{\mathrm e}^{y}}{{\mathrm e}^{y} c_{1} +1}d y &= x +c_{2}\\ \frac {\ln \left ({\mathrm e}^{y} c_{1} +1\right )}{c_{1}}&=x +c_{2} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\ln \left (\frac {{\mathrm e}^{c_{1} c_{2} +c_{1} x}-1}{c_{1}}\right )\\ &=\ln \left (\frac {c_{2} {\mathrm e}^{c_{1} x}-1}{c_{1}}\right ) \end {align*}
Initial conditions are used to solve for the constants of integration.
Looking at the above solution \begin {align*} y = \ln \left (\frac {c_{2} {\mathrm e}^{c_{1} x}-1}{c_{1}}\right ) \tag {1} \end {align*}
Initial conditions are now substituted in the above solution. This will generate the required equations to solve for the integration constants. substituting \(y = 0\) and \(x = 0\) in the above gives \begin {align*} 0 = \ln \left (\frac {c_{2} -1}{c_{1}}\right )\tag {1A} \end {align*}
Taking derivative of the solution gives \begin {align*} y^{\prime } = \frac {c_{2} {\mathrm e}^{c_{1} x} c_{1}}{c_{2} {\mathrm e}^{c_{1} x}-1} \end {align*}
substituting \(y^{\prime } = 2\) and \(x = 0\) in the above gives \begin {align*} 2 = \frac {c_{2} c_{1}}{c_{2} -1}\tag {2A} \end {align*}
Equations {1A,2A} are now solved for \(\{c_{1}, c_{2}\}\). Solving for the constants gives \begin {align*} c_{1}&=1\\ c_{2}&=2 \end {align*}
Substituting these values back in above solution results in \begin {align*} y = \ln \left (2 \,{\mathrm e}^{x}-1\right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \ln \left (2 \,{\mathrm e}^{x}-1\right ) \\ \end{align*}
Verification of solutions
\[ y = \ln \left (2 \,{\mathrm e}^{x}-1\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime \prime }+\frac {y^{\prime }}{{\mathrm e}^{y}}=0, y \left (0\right )=0, y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=2\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & y^{\prime \prime } \\ \bullet & {} & \textrm {Define new dependent variable}\hspace {3pt} u \\ {} & {} & u \left (x \right )=y^{\prime } \\ \bullet & {} & \textrm {Compute}\hspace {3pt} y^{\prime \prime } \\ {} & {} & u^{\prime }\left (x \right )=y^{\prime \prime } \\ \bullet & {} & \textrm {Use chain rule on the lhs}\hspace {3pt} \\ {} & {} & y^{\prime } \left (\frac {d}{d y}u \left (y \right )\right )=y^{\prime \prime } \\ \bullet & {} & \textrm {Substitute in the definition of}\hspace {3pt} u \\ {} & {} & u \left (y \right ) \left (\frac {d}{d y}u \left (y \right )\right )=y^{\prime \prime } \\ \bullet & {} & \textrm {Make substitutions}\hspace {3pt} y^{\prime }=u \left (y \right ),y^{\prime \prime }=u \left (y \right ) \left (\frac {d}{d y}u \left (y \right )\right )\hspace {3pt}\textrm {to reduce order of ODE}\hspace {3pt} \\ {} & {} & u \left (y \right ) \left (\frac {d}{d y}u \left (y \right )\right )+\frac {u \left (y \right )}{{\mathrm e}^{y}}=0 \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d y}u \left (y \right )=-\frac {1}{{\mathrm e}^{y}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} y \\ {} & {} & \int \left (\frac {d}{d y}u \left (y \right )\right )d y =\int -\frac {1}{{\mathrm e}^{y}}d y +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & u \left (y \right )=\frac {1}{{\mathrm e}^{y}}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} u \left (y \right ) \\ {} & {} & u \left (y \right )=\frac {{\mathrm e}^{y} c_{1} +1}{{\mathrm e}^{y}} \\ \bullet & {} & \textrm {Solve 1st ODE for}\hspace {3pt} u \left (y \right ) \\ {} & {} & u \left (y \right )=\frac {{\mathrm e}^{y} c_{1} +1}{{\mathrm e}^{y}} \\ \bullet & {} & \textrm {Revert to original variables with substitution}\hspace {3pt} u \left (y \right )=y^{\prime },y =y \\ {} & {} & y^{\prime }=\frac {{\mathrm e}^{y} c_{1} +1}{{\mathrm e}^{y}} \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {{\mathrm e}^{y} c_{1} +1}{{\mathrm e}^{y}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime } {\mathrm e}^{y}}{{\mathrm e}^{y} c_{1} +1}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime } {\mathrm e}^{y}}{{\mathrm e}^{y} c_{1} +1}d x =\int 1d x +c_{2} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\ln \left ({\mathrm e}^{y} c_{1} +1\right )}{c_{1}}=x +c_{2} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\ln \left (\frac {{\mathrm e}^{c_{2} c_{1} +c_{1} x}-1}{c_{1}}\right ) \\ \square & {} & \textrm {Check validity of solution}\hspace {3pt} y=\ln \left (\frac {{\mathrm e}^{c_{2} c_{1} +c_{1} x}-1}{c_{1}}\right ) \\ {} & \circ & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=0 \\ {} & {} & 0=\ln \left (\frac {{\mathrm e}^{c_{2} c_{1}}-1}{c_{1}}\right ) \\ {} & \circ & \textrm {Compute derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {{\mathrm e}^{c_{2} c_{1} +c_{1} x} c_{1}}{{\mathrm e}^{c_{2} c_{1} +c_{1} x}-1} \\ {} & \circ & \textrm {Use the initial condition}\hspace {3pt} y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=2 \\ {} & {} & 2=\frac {{\mathrm e}^{c_{2} c_{1}} c_{1}}{{\mathrm e}^{c_{2} c_{1}}-1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} c_{1} \hspace {3pt}\textrm {and}\hspace {3pt} c_{2} \\ {} & {} & \left \{c_{1} =1, c_{2} =\ln \left (2\right )\right \} \\ {} & \circ & \textrm {Substitute constant values into general solution and simplify}\hspace {3pt} \\ {} & {} & y=\ln \left (2 \,{\mathrm e}^{x}-1\right ) \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\ln \left (2 \,{\mathrm e}^{x}-1\right ) \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)+_b(_a)*exp(-_a) = 0, _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: 0.407 (sec). Leaf size: 11
dsolve([diff(y(x),x$2)=-diff(y(x),x)*exp(-y(x)),y(0) = 0, D(y)(0) = 2],y(x), singsol=all)
\[ y \left (x \right ) = \ln \left (2 \,{\mathrm e}^{x}-1\right ) \]
✓ Solution by Mathematica
Time used: 5.75 (sec). Leaf size: 13
DSolve[{y''[x]==-y'[x]*Exp[-y[x]],{y[0]==0,y'[0]==2}},y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \log \left (2 e^x-1\right ) \]