Internal problem ID [4049]
Internal file name [OUTPUT/3542_Sunday_June_05_2022_09_36_48_AM_17429707/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 28
Problem number: 810.
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]
\[ \boxed {{y^{\prime }}^{2}+\left (y+x \right ) y^{\prime }+y x=0} \] The ode \begin {align*} {y^{\prime }}^{2}+\left (y+x \right ) y^{\prime }+y x = 0 \end {align*}
is factored to \begin {align*} \left (y^{\prime }+x \right ) \left (y^{\prime }+y\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} y^{\prime }+x = 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 { -x\,\mathop {\mathrm {d}x}}\\ &= -\frac {x^{2}}{2}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {x^{2}}{2}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = -\frac {x^{2}}{2}+c_{1} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {x^{2}}{2}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = -\frac {x^{2}}{2}+c_{1} \] 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} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&={\mathrm e}^{-x -c_{2}}\\ &=\frac {{\mathrm e}^{-x}}{c_{2}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {{\mathrm e}^{-x}}{c_{2}} \\ \end{align*}
Verification of solutions
\[ y = \frac {{\mathrm e}^{-x}}{c_{2}} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {{\mathrm e}^{-x}}{c_{2}} \\ \end{align*}
Verification of solutions
\[ y = \frac {{\mathrm e}^{-x}}{c_{2}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{2}+\left (y+x \right ) y^{\prime }+y x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=-x , y^{\prime }=-y\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-x \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int -x d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\frac {x^{2}}{2}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {x^{2}}{2}+c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-y \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=-1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int \left (-1\right )d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=c_{1} -x \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{c_{1} -x} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=-\frac {x^{2}}{2}+c_{1} , y={\mathrm e}^{c_{1} -x}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature 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: 20
dsolve(diff(y(x),x)^2+(x+y(x))*diff(y(x),x)+x*y(x) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {x^{2}}{2}+c_{1} \\ y \left (x \right ) &= {\mathrm e}^{-x} c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.037 (sec). Leaf size: 32
DSolve[(y'[x])^2+(x+y[x])y'[x]+x y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 e^{-x} \\ y(x)\to -\frac {x^2}{2}+c_1 \\ y(x)\to 0 \\ \end{align*}