Internal problem ID [4221]
Internal file name [OUTPUT/3714_Sunday_June_05_2022_10_23_00_AM_39261125/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 33
Problem number: 989.
ODE order: 1.
ODE degree: 2.
The type(s) of ODE detected by this program : "exact", "differentialType", "homogeneousTypeD2", "first_order_ode_lie_symmetry_calculated"
Maple gives the following as the ode type
[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]
\[ \boxed {\left (y+x \right )^{2} {y^{\prime }}^{2}-\left (x^{2}-y x -2 y^{2}\right ) y^{\prime }-\left (-y+x \right ) y=0} \] The ode \begin {align*} \left (y+x \right )^{2} {y^{\prime }}^{2}-\left (x^{2}-y x -2 y^{2}\right ) y^{\prime }-\left (-y+x \right ) y = 0 \end {align*}
is factored to \begin {align*} \left (y^{\prime } y+x y^{\prime }+y-x \right ) \left (y^{\prime } y+x y^{\prime }+y\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} y^{\prime } y+x y^{\prime }+y-x = 0\tag {1} \\ y^{\prime } y+x y^{\prime }+y = 0\tag {2} \\ \end {align*}
Each of the above equations is now solved.
Solving ODE (1) Using the change of variables \(y = u \left (x \right ) x\) on the above ode results in new ode in \(u \left (x \right )\) \begin {align*} \left (u^{\prime }\left (x \right ) x +u \left (x \right )\right ) u \left (x \right ) x +x \left (u^{\prime }\left (x \right ) x +u \left (x \right )\right )+u \left (x \right ) x = x \end {align*}
In canonical form the ODE is \begin {align*} u' &= F(x,u)\\ &= f( x) g(u)\\ &= -\frac {u^{2}+2 u -1}{x \left (u +1\right )} \end {align*}
Where \(f(x)=-\frac {1}{x}\) and \(g(u)=\frac {u^{2}+2 u -1}{u +1}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {u^{2}+2 u -1}{u +1}} \,du &= -\frac {1}{x} \,d x \\ \int { \frac {1}{\frac {u^{2}+2 u -1}{u +1}} \,du} &= \int {-\frac {1}{x} \,d x} \\ \frac {\ln \left (u^{2}+2 u -1\right )}{2}&=-\ln \left (x \right )+c_{2} \\ \end{align*} Raising both side to exponential gives \begin {align*} \sqrt {u^{2}+2 u -1} &= {\mathrm e}^{-\ln \left (x \right )+c_{2}} \end {align*}
Which simplifies to \begin {align*} \sqrt {u^{2}+2 u -1} &= \frac {c_{3}}{x} \end {align*}
Which simplifies to \[ \sqrt {u \left (x \right )^{2}+2 u \left (x \right )-1} = \frac {c_{3} {\mathrm e}^{c_{2}}}{x} \] The solution is \[ \sqrt {u \left (x \right )^{2}+2 u \left (x \right )-1} = \frac {c_{3} {\mathrm e}^{c_{2}}}{x} \] Replacing \(u(x)\) in the above solution by \(\frac {y}{x}\) results in the solution for \(y\) in implicit form \begin {align*} \sqrt {\frac {y^{2}}{x^{2}}+\frac {2 y}{x}-1} = \frac {c_{3} {\mathrm e}^{c_{2}}}{x}\\ \sqrt {\frac {y^{2}+2 y x -x^{2}}{x^{2}}} = \frac {c_{3} {\mathrm e}^{c_{2}}}{x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \sqrt {\frac {y^{2}+2 y x -x^{2}}{x^{2}}} &= \frac {c_{3} {\mathrm e}^{c_{2}}}{x} \\ \end{align*}
Verification of solutions
\[ \sqrt {\frac {y^{2}+2 y x -x^{2}}{x^{2}}} = \frac {c_{3} {\mathrm e}^{c_{2}}}{x} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} \sqrt {\frac {y^{2}+2 y x -x^{2}}{x^{2}}} &= \frac {c_{3} {\mathrm e}^{c_{2}}}{x} \\ \end{align*}
Verification of solutions
\[ \sqrt {\frac {y^{2}+2 y x -x^{2}}{x^{2}}} = \frac {c_{3} {\mathrm e}^{c_{2}}}{x} \] Verified OK.
Solving ODE (2) Using the change of variables \(y = u \left (x \right ) x\) on the above ode results in new ode in \(u \left (x \right )\) \begin {align*} \left (u^{\prime }\left (x \right ) x +u \left (x \right )\right ) u \left (x \right ) x +x \left (u^{\prime }\left (x \right ) x +u \left (x \right )\right )+u \left (x \right ) x = 0 \end {align*}
In canonical form the ODE is \begin {align*} u' &= F(x,u)\\ &= f( x) g(u)\\ &= -\frac {u \left (u +2\right )}{x \left (u +1\right )} \end {align*}
Where \(f(x)=-\frac {1}{x}\) and \(g(u)=\frac {\left (u +2\right ) u}{u +1}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {\left (u +2\right ) u}{u +1}} \,du &= -\frac {1}{x} \,d x \\ \int { \frac {1}{\frac {\left (u +2\right ) u}{u +1}} \,du} &= \int {-\frac {1}{x} \,d x} \\ \frac {\ln \left (u \left (u +2\right )\right )}{2}&=-\ln \left (x \right )+c_{5} \\ \end{align*} Raising both side to exponential gives \begin {align*} \sqrt {u \left (u +2\right )} &= {\mathrm e}^{-\ln \left (x \right )+c_{5}} \end {align*}
Which simplifies to \begin {align*} \sqrt {u \left (u +2\right )} &= \frac {c_{6}}{x} \end {align*}
Which simplifies to \[ \sqrt {u \left (x \right ) \left (u \left (x \right )+2\right )} = \frac {c_{6} {\mathrm e}^{c_{5}}}{x} \] The solution is \[ \sqrt {u \left (x \right ) \left (u \left (x \right )+2\right )} = \frac {c_{6} {\mathrm e}^{c_{5}}}{x} \] Replacing \(u(x)\) in the above solution by \(\frac {y}{x}\) results in the solution for \(y\) in implicit form \begin {align*} \sqrt {\frac {y \left (\frac {y}{x}+2\right )}{x}} = \frac {c_{6} {\mathrm e}^{c_{5}}}{x}\\ \sqrt {\frac {y \left (2 x +y\right )}{x^{2}}} = \frac {c_{6} {\mathrm e}^{c_{5}}}{x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \sqrt {\frac {y \left (2 x +y\right )}{x^{2}}} &= \frac {c_{6} {\mathrm e}^{c_{5}}}{x} \\ \end{align*}
Verification of solutions
\[ \sqrt {\frac {y \left (2 x +y\right )}{x^{2}}} = \frac {c_{6} {\mathrm e}^{c_{5}}}{x} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} \sqrt {\frac {y \left (2 x +y\right )}{x^{2}}} &= \frac {c_{6} {\mathrm e}^{c_{5}}}{x} \\ \end{align*}
Verification of solutions
\[ \sqrt {\frac {y \left (2 x +y\right )}{x^{2}}} = \frac {c_{6} {\mathrm e}^{c_{5}}}{x} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (y+x \right )^{2} {y^{\prime }}^{2}-\left (x^{2}-y x -2 y^{2}\right ) y^{\prime }-\left (-y+x \right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=-\frac {y}{y+x}, y^{\prime }=-\frac {y-x}{y+x}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {y}{y+x} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {y-x}{y+x} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\mathit {workingODE} , \mathit {workingODE}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear <- 1st order linear successful <- inverse linear successful Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying homogeneous D <- homogeneous successful`
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 85
dsolve((x+y(x))^2*diff(y(x),x)^2-(x^2-x*y(x)-2*y(x)^2)*diff(y(x),x)-(x-y(x))*y(x) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -x -\sqrt {x^{2}+2 c_{1}} \\ y \left (x \right ) &= -x +\sqrt {x^{2}+2 c_{1}} \\ y \left (x \right ) &= \frac {-c_{1} x -\sqrt {2 c_{1}^{2} x^{2}+1}}{c_{1}} \\ y \left (x \right ) &= \frac {-c_{1} x +\sqrt {2 c_{1}^{2} x^{2}+1}}{c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.592 (sec). Leaf size: 172
DSolve[(x+y[x])^2 (y'[x])^2 -(x^2-x y[x]-2 y[x]^2) y'[x]-(x-y[x])y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -x-\sqrt {x^2+e^{2 c_1}} \\ y(x)\to -x+\sqrt {x^2+e^{2 c_1}} \\ y(x)\to -x-\sqrt {2 x^2+e^{2 c_1}} \\ y(x)\to -x+\sqrt {2 x^2+e^{2 c_1}} \\ y(x)\to -\sqrt {x^2}-x \\ y(x)\to \sqrt {x^2}-x \\ y(x)\to -\sqrt {2} \sqrt {x^2}-x \\ y(x)\to \sqrt {2} \sqrt {x^2}-x \\ \end{align*}