Internal problem ID [3200]
Internal file name [OUTPUT/2692_Sunday_June_05_2022_08_38_53_AM_37364604/index.tex]
Book: Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section: Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page
78
Problem number: 56.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {-\left (y-2 x y\right ) y^{\prime }=-1} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= -\frac {1}{y \left (2 x -1\right )} \end {align*}
Where \(f(x)=-\frac {1}{2 x -1}\) and \(g(y)=\frac {1}{y}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {1}{y}} \,dy &= -\frac {1}{2 x -1} \,d x \\ \int { \frac {1}{\frac {1}{y}} \,dy} &= \int {-\frac {1}{2 x -1} \,d x} \\ \frac {y^{2}}{2}&=-\frac {\ln \left (2 x -1\right )}{2}+c_{1} \\ \end{align*} Which results in \begin{align*} y &= \sqrt {-\ln \left (2 x -1\right )+2 c_{1}} \\ y &= -\sqrt {-\ln \left (2 x -1\right )+2 c_{1}} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \sqrt {-\ln \left (2 x -1\right )+2 c_{1}} \\ \tag{2} y &= -\sqrt {-\ln \left (2 x -1\right )+2 c_{1}} \\ \end{align*}
Verification of solutions
\[ y = \sqrt {-\ln \left (2 x -1\right )+2 c_{1}} \] Verified OK.
\[ y = -\sqrt {-\ln \left (2 x -1\right )+2 c_{1}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -\left (y-2 x y\right ) y^{\prime }=-1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {1}{y-2 x y} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y y^{\prime }=-\frac {1}{2 x -1} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y y^{\prime }d x =\int -\frac {1}{2 x -1}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {y^{2}}{2}=-\frac {\ln \left (2 x -1\right )}{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\sqrt {-\ln \left (2 x -1\right )+2 c_{1}}, y=-\sqrt {-\ln \left (2 x -1\right )+2 c_{1}}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 33
dsolve(1-(y(x)-2*x*y(x))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \sqrt {-\ln \left (2 x -1\right )+c_{1}} \\ y \left (x \right ) &= -\sqrt {-\ln \left (2 x -1\right )+c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.013 (sec). Leaf size: 45
DSolve[1-(y[x]-2*x*y[x])*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {-\log (1-2 x)+2 c_1} \\ y(x)\to \sqrt {-\log (1-2 x)+2 c_1} \\ \end{align*}