Internal problem ID [4185]
Internal file name [OUTPUT/3678_Sunday_June_05_2022_10_08_52_AM_945372/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 32
Problem number: 951.
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]
\[ \boxed {y {y^{\prime }}^{2}-\left (y x +1\right ) y^{\prime }=-x} \] The ode \begin {align*} y {y^{\prime }}^{2}-\left (y x +1\right ) y^{\prime } = -x \end {align*}
is factored to \begin {align*} \left (y^{\prime }-x \right ) \left (y^{\prime } y-1\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} y^{\prime }-x = 0\tag {1} \\ y^{\prime } y-1 = 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 y d y &= x +c_{2}\\ \frac {y^{2}}{2}&=x +c_{2} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\sqrt {2 x +2 c_{2}}\\ y_2&=-\sqrt {2 x +2 c_{2}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \sqrt {2 x +2 c_{2}} \\ \tag{2} y &= -\sqrt {2 x +2 c_{2}} \\ \end{align*}
Verification of solutions
\[ y = \sqrt {2 x +2 c_{2}} \] Verified OK.
\[ y = -\sqrt {2 x +2 c_{2}} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \sqrt {2 x +2 c_{2}} \\ \tag{2} y &= -\sqrt {2 x +2 c_{2}} \\ \end{align*}
Verification of solutions
\[ y = \sqrt {2 x +2 c_{2}} \] Verified OK.
\[ y = -\sqrt {2 x +2 c_{2}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y {y^{\prime }}^{2}-\left (y x +1\right ) y^{\prime }=-x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=x , y^{\prime }=\frac {1}{y}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=x \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int x d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {x^{2}}{2}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {x^{2}}{2}+c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {1}{y} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime } y=1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime } yd x =\int 1d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {y^{2}}{2}=x +c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\sqrt {2 x +2 c_{1}}, y=-\sqrt {2 x +2 c_{1}}\right \} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=\frac {x^{2}}{2}+c_{1} , \left \{y=\sqrt {2 x +2 c_{1}}, y=-\sqrt {2 x +2 c_{1}}\right \}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 33
dsolve(y(x)*diff(y(x),x)^2-(1+x*y(x))*diff(y(x),x)+x = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \sqrt {2 x +c_{1}} \\ y \left (x \right ) &= -\sqrt {2 x +c_{1}} \\ y \left (x \right ) &= \frac {x^{2}}{2}+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.1 (sec). Leaf size: 52
DSolve[y[x] (y'[x])^2-(1+x y[x])y'[x]+x==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {2} \sqrt {x+c_1} \\ y(x)\to \sqrt {2} \sqrt {x+c_1} \\ y(x)\to \frac {x^2}{2}+c_1 \\ \end{align*}