Internal problem ID [15002]
Internal file name [OUTPUT/15003_Friday_April_19_2024_04_43_46_AM_59032490/index.tex]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Section 4. Equations with variables separable and equations reducible to them. Exercises
page 38
Problem number: 96.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {{\mathrm e}^{y}-{\mathrm e}^{4 y} y^{\prime }=1} \]
Integrating both sides gives \begin {align*} \int \frac {{\mathrm e}^{4 y}}{-1+{\mathrm e}^{y}}d y &= \int {dx}\\ \int _{}^{y}\frac {{\mathrm e}^{4 \textit {\_a}}}{-1+{\mathrm e}^{\textit {\_a}}}d \textit {\_a}&= x +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {{\mathrm e}^{4 \textit {\_a}}}{-1+{\mathrm e}^{\textit {\_a}}}d \textit {\_a} &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}\frac {{\mathrm e}^{4 \textit {\_a}}}{-1+{\mathrm e}^{\textit {\_a}}}d \textit {\_a} = x +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {\mathrm e}^{y}-{\mathrm e}^{4 y} y^{\prime }=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-{\mathrm e}^{y}+1}{{\mathrm e}^{4 y}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime } {\mathrm e}^{4 y}}{-{\mathrm e}^{y}+1}=-1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime } {\mathrm e}^{4 y}}{-{\mathrm e}^{y}+1}d x =\int \left (-1\right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {\left ({\mathrm e}^{y}\right )^{3}}{3}-\frac {\left ({\mathrm e}^{y}\right )^{2}}{2}-{\mathrm e}^{y}-\ln \left ({\mathrm e}^{y}-1\right )=-x +c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 33
dsolve(exp(y(x))=exp(4*y(x))*diff(y(x),x)+1,y(x), singsol=all)
\[ x -\frac {{\mathrm e}^{3 y}}{3}-\frac {{\mathrm e}^{2 y}}{2}-{\mathrm e}^{y}-\ln \left ({\mathrm e}^{y}-1\right )+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.366 (sec). Leaf size: 48
DSolve[Exp[y[x]]==Exp[4*y[x]]*y'[x]+1,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {1}{6} e^{\text {$\#$1}} \left (3 e^{\text {$\#$1}}+2 e^{2 \text {$\#$1}}+6\right )+\log \left (e^{\text {$\#$1}}-1\right )\&\right ][x+c_1] \\ y(x)\to 0 \\ \end{align*}