Internal problem ID [11199]
Internal file name [OUTPUT/10185_Tuesday_December_06_2022_03_59_36_AM_9550685/index.tex
]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IV, differential equations of the first order and higher degree than the first.
Article 24. Equations solvable for \(p\). Page 49
Problem number: Ex 1.
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]
\[ \boxed {{y^{\prime }}^{2}+\left (y+x \right ) y^{\prime }+x y=0} \] The ode \begin {align*} {y^{\prime }}^{2}+\left (y+x \right ) y^{\prime }+x y = 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.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{2}+\left (y+x \right ) y^{\prime }+x y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=-x , y^{\prime }=-y\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-x \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int -x d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\frac {x^{2}}{2}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {x^{2}}{2}+c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-y \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=-1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int \left (-1\right )d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=-x +c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{-x +c_{1}} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=-\frac {x^{2}}{2}+c_{1} , y={\mathrm e}^{-x +c_{1}}\right \} \end {array} \]
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 ) &= c_{1} {\mathrm e}^{-x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.056 (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*}