Internal problem ID [5748]
Internal file name [OUTPUT/4996_Sunday_June_05_2022_03_16_38_PM_17586316/index.tex]
Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold
Scientific. Singapore. 1995
Section: Chapter 1. First order differential equations. Section 1.1 Separable equations problems.
page 7
Problem number: 35.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {\left (1+y^{2}\right ) \left ({\mathrm e}^{2 x}-y^{\prime } {\mathrm e}^{y}\right )-\left (y+1\right ) y^{\prime }=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {{\mathrm e}^{2 x} \left (y^{2}+1\right )}{{\mathrm e}^{y} y^{2}+{\mathrm e}^{y}+y +1} \end {align*}
Where \(f(x)={\mathrm e}^{2 x}\) and \(g(y)=\frac {y^{2}+1}{{\mathrm e}^{y} y^{2}+{\mathrm e}^{y}+y +1}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {y^{2}+1}{{\mathrm e}^{y} y^{2}+{\mathrm e}^{y}+y +1}} \,dy &= {\mathrm e}^{2 x} \,d x \\ \int { \frac {1}{\frac {y^{2}+1}{{\mathrm e}^{y} y^{2}+{\mathrm e}^{y}+y +1}} \,dy} &= \int {{\mathrm e}^{2 x} \,d x} \\ \arctan \left (y \right )+\frac {\ln \left (y^{2}+1\right )}{2}+{\mathrm e}^{y}&=\frac {{\mathrm e}^{2 x}}{2}+c_{1} \\ \end{align*} The solution is \[ \arctan \left (y\right )+\frac {\ln \left (1+y^{2}\right )}{2}+{\mathrm e}^{y}-\frac {{\mathrm e}^{2 x}}{2}-c_{1} = 0 \]
The solution(s) found are the following \begin{align*} \tag{1} \arctan \left (y\right )+\frac {\ln \left (1+y^{2}\right )}{2}+{\mathrm e}^{y}-\frac {{\mathrm e}^{2 x}}{2}-c_{1} &= 0 \\ \end{align*}
Verification of solutions
\[ \arctan \left (y\right )+\frac {\ln \left (1+y^{2}\right )}{2}+{\mathrm e}^{y}-\frac {{\mathrm e}^{2 x}}{2}-c_{1} = 0 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (1+y^{2}\right ) \left ({\mathrm e}^{2 x}-y^{\prime } {\mathrm e}^{y}\right )-\left (y+1\right ) y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {\left (1+y^{2}\right ) {\mathrm e}^{2 x}}{-\left (1+y^{2}\right ) {\mathrm e}^{y}-y-1} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime } \left (-\left (1+y^{2}\right ) {\mathrm e}^{y}-y-1\right )}{1+y^{2}}=-{\mathrm e}^{2 x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime } \left (-\left (1+y^{2}\right ) {\mathrm e}^{y}-y-1\right )}{1+y^{2}}d x =\int -{\mathrm e}^{2 x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\arctan \left (y\right )-\frac {\ln \left (1+y^{2}\right )}{2}-{\mathrm e}^{y}=-\frac {{\mathrm e}^{2 x}}{2}+c_{1} \\ \bullet & {} & \textrm {Convert}\hspace {3pt} \arctan \mathrm {to \esapos ln\esapos } \\ {} & {} & -\frac {\mathrm {I} \left (\ln \left (1-\mathrm {I} y\right )-\ln \left (1+\mathrm {I} y\right )\right )}{2}-\frac {\ln \left (1+y^{2}\right )}{2}-{\mathrm e}^{y}=-\frac {{\mathrm e}^{2 x}}{2}+c_{1} \end {array} \]
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: 0.093 (sec). Leaf size: 30
dsolve((1+y(x)^2)*(exp(2*x)-exp(y(x))*diff(y(x),x))-(1+y(x))*diff(y(x),x)=0,y(x), singsol=all)
\[ \frac {{\mathrm e}^{2 x}}{2}-\arctan \left (y \left (x \right )\right )-\frac {\ln \left (1+y \left (x \right )^{2}\right )}{2}-{\mathrm e}^{y \left (x \right )}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.696 (sec). Leaf size: 70
DSolve[(1+y[x]^2)*(Exp[2*x]-Exp[y[x]]*y'[x])-(1+y[x])*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [e^{\text {$\#$1}}+\left (\frac {1}{2}-\frac {i}{2}\right ) \log (-\text {$\#$1}+i)+\left (\frac {1}{2}+\frac {i}{2}\right ) \log (\text {$\#$1}+i)\&\right ]\left [\frac {e^{2 x}}{2}+c_1\right ] \\ y(x)\to -i \\ y(x)\to i \\ \end{align*}