Internal problem ID [14951]
Internal file name [OUTPUT/14961_Monday_April_15_2024_12_04_23_AM_41410347/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 2. The method of isoclines. Exercises page 27
Problem number: 25.
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]
\[ \boxed {y^{\prime }-\left (y-1\right )^{2}=0} \]
Integrating both sides gives \begin {align*} \int \frac {1}{\left (y -1\right )^{2}}d y &= x +c_{1}\\ -\frac {1}{y -1}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {x +c_{1} -1}{x +c_{1}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x +c_{1} -1}{x +c_{1}} \\ \end{align*}
Verification of solutions
\[ y = \frac {x +c_{1} -1}{x +c_{1}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\left (y-1\right )^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\left (y-1\right )^{2} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\left (y-1\right )^{2}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\left (y-1\right )^{2}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{y-1}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {x +c_{1} -1}{x +c_{1}} \end {array} \]
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: 0.0 (sec). Leaf size: 14
dsolve(diff(y(x),x)=(y(x)-1)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} +x -1}{c_{1} +x} \]
✓ Solution by Mathematica
Time used: 0.107 (sec). Leaf size: 22
DSolve[y'[x]==(y[x]-1)^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x-1+c_1}{x+c_1} \\ y(x)\to 1 \\ \end{align*}