Internal problem ID [1034]
Internal file name [OUTPUT/1035_Sunday_June_05_2022_01_57_26_AM_68709905/index.tex]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 2, First order equations. Exact equations. Section 2.5 Page 79
Problem number: 5.
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]
\[ \boxed {\left (x +y\right )^{2}+\left (x +y\right )^{2} y^{\prime }=0} \]
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.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (x +y\right )^{2}+\left (x +y\right )^{2} y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime }=-1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \left (-1\right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-x +c_{1} \end {array} \]
Maple trace
`Classification methods on request Methods to be used are: [exact] ---------------------------- * Tackling ODE using method: exact --- Trying classification methods --- trying exact <- exact successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 49
dsolve((x+y(x))^2+(x+y(x))^2*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -x \\ y \left (x \right ) &= c_{1} -x \\ y \left (x \right ) &= -\frac {c_{1}}{2}-\frac {i \sqrt {3}\, c_{1}}{2}-x \\ y \left (x \right ) &= -\frac {c_{1}}{2}+\frac {i \sqrt {3}\, c_{1}}{2}-x \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 18
DSolve[(x+y[x])^2+(x+y[x])^2*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -x \\ y(x)\to -x+c_1 \\ \end{align*}