Internal problem ID [8349]
Internal file name [OUTPUT/7282_Sunday_June_05_2022_05_42_19_PM_69996562/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 12.
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]
\[ \boxed {y^{\prime }+y^{2}=1} \]
Integrating both sides gives \begin {align*} \int \frac {1}{-y^{2}+1}d y &= x +c_{1}\\ \operatorname {arctanh}\left (y \right )&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\tanh \left (x +c_{1} \right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \tanh \left (x +c_{1} \right ) \\ \end{align*}
Verification of solutions
\[ y = \tanh \left (x +c_{1} \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+y^{2}=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-y^{2}+1 \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-y^{2}+1}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-y^{2}+1}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \mathrm {arctanh}\left (y\right )=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\tanh \left (x +c_{1} \right ) \end {array} \]
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: 0.0 (sec). Leaf size: 8
dsolve(diff(y(x),x) + y(x)^2 - 1=0,y(x), singsol=all)
\[ y \left (x \right ) = \tanh \left (x +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.638 (sec). Leaf size: 44
DSolve[y'[x] + y[x]^2 - 1==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{2 x}-e^{2 c_1}}{e^{2 x}+e^{2 c_1}} \\ y(x)\to -1 \\ y(x)\to 1 \\ \end{align*}