Internal problem ID [12482]
Internal file name [OUTPUT/11135_Monday_October_16_2023_09_51_43_PM_60606324/index.tex]
Book: DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR
PUBLISHERS, Moscow 1969.
Section: Chapter 8. Differential equations. Exercises page 595
Problem number: 94.
ODE order: 1.
ODE degree: 2.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {y-y y^{\prime }-y^{\prime }+{y^{\prime }}^{2}=0} \] The ode \begin {align*} y-y y^{\prime }-y^{\prime }+{y^{\prime }}^{2} = 0 \end {align*}
is factored to \begin {align*} \left (y^{\prime }-1\right ) \left (y^{\prime }-y\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} y^{\prime }-1 = 0\tag {1} \\ y^{\prime }-y = 0\tag {2} \\ \end {align*}
Each of the above equations is now solved.
Solving ODE (1) Integrating both sides gives \begin {align*} y &= \int { 1\,\mathop {\mathrm {d}x}}\\ &= x +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ y = x +c_{1} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ y = x +c_{1} \] Verified OK.
Solving ODE (2) Integrating both sides gives \begin {align*} \int \frac {1}{y}d y &= x +c_{2}\\ \ln \left (y \right )&=x +c_{2}\\ y&={\mathrm e}^{x +c_{2}}\\ y&=c_{2} {\mathrm e}^{x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{2} {\mathrm e}^{x} \\ \end{align*}
Verification of solutions
\[ y = c_{2} {\mathrm e}^{x} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{2} {\mathrm e}^{x} \\ \end{align*}
Verification of solutions
\[ y = c_{2} {\mathrm e}^{x} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y-y y^{\prime }-y^{\prime }+{y^{\prime }}^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=1, y^{\prime }=y\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int 1d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=x +c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=x +c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=y \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int 1d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=x +c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{x +c_{1}} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=x +c_{1} , y={\mathrm e}^{x +c_{1}}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 14
dsolve(y(x)=y(x)*diff(y(x),x)+diff(y(x),x)-diff(y(x),x)^2,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= c_{1} +x \\ y \left (x \right ) &= c_{1} {\mathrm e}^{x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.019 (sec). Leaf size: 19
DSolve[y[x]==y[x]*y'[x]+y'[x]-(y'[x])^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 e^x \\ y(x)\to x+c_1 \\ \end{align*}