Internal problem ID [7068]
Internal file name [OUTPUT/6054_Sunday_June_05_2022_04_15_43_PM_10654432/index.tex]
Book: Own collection of miscellaneous problems
Section: section 1.0
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]
\[ \boxed {\left (x +y\right ) y^{\prime }=0} \] The ode \begin {align*} \left (x +y\right ) y^{\prime } = 0 \end {align*}
Gives the following equations \begin {align*} x +y = 0\tag {1} \\ y^{\prime } = 0\tag {2} \\ \end {align*}
Each of the above equations is now solved.
Solving ODE (1) Since \(y = -x\), is missing derivative in \(y\) then it is an algebraic equation. Solving for \(y\). \begin {align*} \end {align*}
Solving ODE (2) Integrating both sides gives \begin {align*} y &= \int { 0\,\mathop {\mathrm {d}x}}\\ &= c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \\ \end{align*}
Verification of solutions
\[ y = c_{1} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \\ \end{align*}
Verification of solutions
\[ y = c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (x +y\right ) y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=0 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int 0d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=c_{1} \end {array} \]
Maple trace
`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: 11
dsolve((x+y(x))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -x \\ y \left (x \right ) &= c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 14
DSolve[(x+y[x])*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -x \\ y(x)\to c_1 \\ \end{align*}