Internal problem ID [6092]
Internal file name [OUTPUT/5340_Sunday_June_05_2022_03_34_43_PM_40990985/index.tex]
Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY
1961
Section: Chapter 6. Existence and uniqueness of solutions to systems and nth order equations.
Page 238
Problem number: 1(b).
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]]
\[ \boxed {y^{\prime \prime }+y^{\prime } {\mathrm e}^{x}={\mathrm e}^{x}} \]
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 )-{\mathrm e}^{x} = 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} \left (-p +1\right ) \end {align*}
Where \(f(x)={\mathrm e}^{x}\) and \(g(p)=-p +1\). Integrating both sides gives \begin{align*} \frac {1}{-p +1} \,dp &= {\mathrm e}^{x} \,d x \\ \int { \frac {1}{-p +1} \,dp} &= \int {{\mathrm e}^{x} \,d x} \\ -\ln \left (p -1\right )&={\mathrm e}^{x}+c_{1} \\ \end{align*} Raising both side to exponential gives \begin {align*} \frac {1}{p -1} &= {\mathrm e}^{{\mathrm e}^{x}+c_{1}} \end {align*}
Which simplifies to \begin {align*} \frac {1}{p -1} &= c_{2} {\mathrm e}^{{\mathrm e}^{x}} \end {align*}
Since \(p=y^{\prime }\) then the new first order ode to solve is \begin {align*} y^{\prime } = \frac {\left (c_{2} {\mathrm e}^{{\mathrm e}^{x}+c_{1}}+1\right ) {\mathrm e}^{-{\mathrm e}^{x}-c_{1}}}{c_{2}} \end {align*}
Integrating both sides gives \begin {align*} y &= \int { \frac {\left (c_{2} {\mathrm e}^{{\mathrm e}^{x}+c_{1}}+1\right ) {\mathrm e}^{-{\mathrm e}^{x}-c_{1}}}{c_{2}}\,\mathop {\mathrm {d}x}}\\ &= x -\frac {{\mathrm e}^{-c_{1}} \operatorname {expIntegral}_{1}\left ({\mathrm e}^{x}\right )}{c_{2}}+c_{3} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= x -\frac {{\mathrm e}^{-c_{1}} \operatorname {expIntegral}_{1}\left ({\mathrm e}^{x}\right )}{c_{2}}+c_{3} \\ \end{align*}
Verification of solutions
\[ y = x -\frac {{\mathrm e}^{-c_{1}} \operatorname {expIntegral}_{1}\left ({\mathrm e}^{x}\right )}{c_{2}}+c_{3} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime }+y^{\prime } {\mathrm e}^{x}={\mathrm e}^{x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & y^{\prime \prime } \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} u =y^{\prime }\hspace {3pt}\textrm {to reduce order of ODE}\hspace {3pt} \\ {} & {} & u^{\prime }\left (x \right )+u \left (x \right ) {\mathrm e}^{x}={\mathrm e}^{x} \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & u^{\prime }\left (x \right )=-u \left (x \right ) {\mathrm e}^{x}+{\mathrm e}^{x} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {u^{\prime }\left (x \right )}{-1+u \left (x \right )}=-{\mathrm e}^{x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {u^{\prime }\left (x \right )}{-1+u \left (x \right )}d x =\int -{\mathrm e}^{x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (-1+u \left (x \right )\right )=-{\mathrm e}^{x}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} u \left (x \right ) \\ {} & {} & u \left (x \right )={\mathrm e}^{-{\mathrm e}^{x}+c_{1}}+1 \\ \bullet & {} & \textrm {Solve 1st ODE for}\hspace {3pt} u \left (x \right ) \\ {} & {} & u \left (x \right )={\mathrm e}^{-{\mathrm e}^{x}+c_{1}}+1 \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} u =y^{\prime } \\ {} & {} & y^{\prime }={\mathrm e}^{-{\mathrm e}^{x}+c_{1}}+1 \\ \bullet & {} & \textrm {Integrate both sides to solve for}\hspace {3pt} y \\ {} & {} & \int y^{\prime }d x =\int \left ({\mathrm e}^{-{\mathrm e}^{x}+c_{1}}+1\right )d x +c_{2} \\ \bullet & {} & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y=x -{\mathrm e}^{c_{1}} \mathrm {Ei}_{1}\left ({\mathrm e}^{x}\right )+c_{2} \end {array} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable -> Calling odsolve with the ODE`, diff(_b(_a), _a) = -exp(_a)*_b(_a)+exp(_a), _b(_a)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful <- high order exact linear fully integrable successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 14
dsolve(diff(y(x),x$2)+exp(x)*diff(y(x),x)=exp(x),y(x), singsol=all)
\[ y \left (x \right ) = -c_{1} \operatorname {expIntegral}_{1}\left ({\mathrm e}^{x}\right )+x +c_{2} \]
✓ Solution by Mathematica
Time used: 0.081 (sec). Leaf size: 18
DSolve[y''[x]+Exp[x]*y'[x]==Exp[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \operatorname {ExpIntegralEi}\left (-e^x\right )+x+c_2 \]