Internal problem ID [3773]
Internal file name [OUTPUT/3266_Sunday_June_05_2022_09_04_26_AM_5486960/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 19
Problem number: 521.
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]
\[ \boxed {x \left (2+y\right ) y^{\prime }=-x a} \]
Integrating both sides gives \begin {align*} \int -\frac {2+y}{a}d y &= x +c_{1}\\ -\frac {y \left (4+y \right )}{2 a}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=-2-\sqrt {-2 a c_{1} -2 x a +4}\\ y_2&=-2+\sqrt {-2 a c_{1} -2 x a +4} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -2-\sqrt {-2 a c_{1} -2 x a +4} \\ \tag{2} y &= -2+\sqrt {-2 a c_{1} -2 x a +4} \\ \end{align*}
Verification of solutions
\[ y = -2-\sqrt {-2 a c_{1} -2 x a +4} \] Verified OK.
\[ y = -2+\sqrt {-2 a c_{1} -2 x a +4} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x \left (2+y\right ) y^{\prime }=-x a \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {a}{2+y} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime } \left (2+y\right )=-a \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime } \left (2+y\right )d x =\int -a d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {y^{2}}{2}+2 y=-x a +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=-2-\sqrt {-2 x a +2 c_{1} +4}, y=-2+\sqrt {-2 x a +2 c_{1} +4}\right \} \end {array} \]
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: 0.015 (sec). Leaf size: 39
dsolve(x*(2+y(x))*diff(y(x),x)+a*x = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -2-\sqrt {4+\left (-2 x -2 c_{1} \right ) a} \\ y \left (x \right ) &= -2+\sqrt {4+\left (-2 x -2 c_{1} \right ) a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.161 (sec). Leaf size: 50
DSolve[x(2+y[x])y'[x]+a x==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -2-\sqrt {2} \sqrt {-a x+2+c_1} \\ y(x)\to -2+\sqrt {2} \sqrt {-a x+2+c_1} \\ \end{align*}