Internal problem ID [2352]
Internal file name [OUTPUT/2352_Tuesday_February_27_2024_08_34_16_AM_70630014/index.tex]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 38, page 173
Problem number: 15.
ODE order: 1.
ODE degree: 3.
The type(s) of ODE detected by this program : "dAlembert"
Maple gives the following as the ode type
[[_1st_order, _with_linear_symmetries], _dAlembert]
\[ \boxed {\frac {1}{{y^{\prime }}^{2}}+y^{\prime } x -2 y=0} \]
Let \(p=y^{\prime }\) the ode becomes \begin {align*} -p^{3} x +2 p^{2} y -1 = 0 \end {align*}
Solving for \(y\) from the above results in \begin {align*} y &= \frac {p x}{2}+\frac {1}{2 p^{2}}\tag {1A} \end {align*}
This has the form \begin {align*} y=xf(p)+g(p)\tag {*} \end {align*}
Where \(f,g\) are functions of \(p=y'(x)\). The above ode is dAlembert ode which is now solved. Taking derivative of (*) w.r.t. \(x\) gives \begin {align*} p &= f+(x f'+g') \frac {dp}{dx}\\ p-f &= (x f'+g') \frac {dp}{dx}\tag {2} \end {align*}
Comparing the form \(y=x f + g\) to (1A) shows that \begin {align*} f &= \frac {p}{2}\\ g &= \frac {1}{2 p^{2}} \end {align*}
Hence (2) becomes \begin {align*} \frac {p}{2} = \left (\frac {x}{2}-\frac {1}{p^{3}}\right ) p^{\prime }\left (x \right )\tag {2A} \end {align*}
The singular solution is found by setting \(\frac {dp}{dx}=0\) in the above which gives \begin {align*} \frac {p}{2} = 0 \end {align*}
Solving for \(p\) from the above gives \begin {align*} p&=0 \end {align*}
Substituting these in (1A) gives \begin {align*} y&=\infty \end {align*}
The general solution is found when \( \frac { \mathop {\mathrm {d}p}}{\mathop {\mathrm {d}x}}\neq 0\). From eq. (2A). This results in \begin {align*} p^{\prime }\left (x \right ) = \frac {p \left (x \right )}{x -\frac {2}{p \left (x \right )^{3}}}\tag {3} \end {align*}
This ODE is now solved for \(p \left (x \right )\).
Inverting the above ode gives \begin {align*} \frac {d}{d p}x \left (p \right ) = \frac {x \left (p \right )-\frac {2}{p^{3}}}{p}\tag {4} \end {align*}
This ODE is now solved for \(x \left (p \right )\).
Entering Linear first order ODE solver. In canonical form a linear first order is \begin {align*} \frac {d}{d p}x \left (p \right ) + p(p)x \left (p \right ) &= q(p) \end {align*}
Where here \begin {align*} p(p) &=-\frac {1}{p}\\ q(p) &=-\frac {2}{p^{4}} \end {align*}
Hence the ode is \begin {align*} \frac {d}{d p}x \left (p \right )-\frac {x \left (p \right )}{p} = -\frac {2}{p^{4}} \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int -\frac {1}{p}d p} \\ &= \frac {1}{p} \\ \end{align*} The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}p}}\left ( \mu x\right ) &= \left (\mu \right ) \left (-\frac {2}{p^{4}}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}p}} \left (\frac {x}{p}\right ) &= \left (\frac {1}{p}\right ) \left (-\frac {2}{p^{4}}\right )\\ \mathrm {d} \left (\frac {x}{p}\right ) &= \left (-\frac {2}{p^{5}}\right )\, \mathrm {d} p \end {align*}
Integrating gives \begin {align*} \frac {x}{p} &= \int {-\frac {2}{p^{5}}\,\mathrm {d} p}\\ \frac {x}{p} &= \frac {1}{2 p^{4}} + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu =\frac {1}{p}\) results in \begin {align*} x \left (p \right ) &= \frac {1}{2 p^{3}}+c_{1} p \end {align*}
Now we need to eliminate \(p\) between the above and (1A). One way to do this is by solving (1) for \(p\). This results in \begin {align*} p&=\frac {\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}}{6 x}+\frac {8 y^{2}}{3 x \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}}+\frac {2 y}{3 x}\\ p&=-\frac {\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}}{12 x}-\frac {4 y^{2}}{3 x \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}}+\frac {2 y}{3 x}+\frac {i \sqrt {3}\, \left (\frac {\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}}{6 x}-\frac {8 y^{2}}{3 x \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}}\right )}{2}\\ p&=-\frac {\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}}{12 x}-\frac {4 y^{2}}{3 x \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}}+\frac {2 y}{3 x}-\frac {i \sqrt {3}\, \left (\frac {\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}}{6 x}-\frac {8 y^{2}}{3 x \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}
Substituting the above in the solution for \(x\) found above gives \begin{align*} x&=\frac {108 x^{3}}{\left (\frac {16 y^{2}}{\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}}+4 y+\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}\right )^{3}}+\frac {c_{1} \left (\frac {16 y^{2}}{\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}}+4 y+\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}\right )}{6 x} \\ x&=\frac {-10368 x^{4} \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}-55296 y^{3} x^{3}+93312 x^{5}}{{\left (-i \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}} \sqrt {3}+16 i \sqrt {3}\, y^{2}+\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}}-8 y \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}+16 y^{2}\right )}^{3}}+\frac {c_{1} \left (\left (i \sqrt {3}-1\right ) \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}}+8 y \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}-16 \left (1+i \sqrt {3}\right ) y^{2}\right )}{12 \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}} x} \\ x&=\frac {3456 \left (-3 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x -16 y^{3}+27 x^{2}\right ) x^{3}}{{\left (i \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}} \sqrt {3}-16 i \sqrt {3}\, y^{2}+\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}}-8 y \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}+16 y^{2}\right )}^{3}}-\frac {c_{1} \left (\left (1+i \sqrt {3}\right ) \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}}-8 y \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}-16 y^{2} \left (i \sqrt {3}-1\right )\right )}{12 \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}} x} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \infty \\ \tag{2} x &= \frac {108 x^{3}}{\left (\frac {16 y^{2}}{\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}}+4 y+\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}\right )^{3}}+\frac {c_{1} \left (\frac {16 y^{2}}{\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}}+4 y+\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}\right )}{6 x} \\ \tag{3} x &= \frac {-10368 x^{4} \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}-55296 y^{3} x^{3}+93312 x^{5}}{{\left (-i \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}} \sqrt {3}+16 i \sqrt {3}\, y^{2}+\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}}-8 y \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}+16 y^{2}\right )}^{3}}+\frac {c_{1} \left (\left (i \sqrt {3}-1\right ) \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}}+8 y \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}-16 \left (1+i \sqrt {3}\right ) y^{2}\right )}{12 \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}} x} \\ \tag{4} x &= \frac {3456 \left (-3 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x -16 y^{3}+27 x^{2}\right ) x^{3}}{{\left (i \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}} \sqrt {3}-16 i \sqrt {3}\, y^{2}+\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}}-8 y \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}+16 y^{2}\right )}^{3}}-\frac {c_{1} \left (\left (1+i \sqrt {3}\right ) \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}}-8 y \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}-16 y^{2} \left (i \sqrt {3}-1\right )\right )}{12 \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}} x} \\ \end{align*}
Verification of solutions
\[ y = \infty \] Warning, solution could not be verified
\[ x = \frac {108 x^{3}}{\left (\frac {16 y^{2}}{\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}}+4 y+\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}\right )^{3}}+\frac {c_{1} \left (\frac {16 y^{2}}{\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}}+4 y+\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}\right )}{6 x} \] Verified OK.
\[ x = \frac {-10368 x^{4} \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}-55296 y^{3} x^{3}+93312 x^{5}}{{\left (-i \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}} \sqrt {3}+16 i \sqrt {3}\, y^{2}+\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}}-8 y \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}+16 y^{2}\right )}^{3}}+\frac {c_{1} \left (\left (i \sqrt {3}-1\right ) \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}}+8 y \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}-16 \left (1+i \sqrt {3}\right ) y^{2}\right )}{12 \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}} x} \] Warning, solution could not be verified
\[ x = \frac {3456 \left (-3 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x -16 y^{3}+27 x^{2}\right ) x^{3}}{{\left (i \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}} \sqrt {3}-16 i \sqrt {3}\, y^{2}+\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}}-8 y \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}+16 y^{2}\right )}^{3}}-\frac {c_{1} \left (\left (1+i \sqrt {3}\right ) \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}}-8 y \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}-16 y^{2} \left (i \sqrt {3}-1\right )\right )}{12 \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}} x} \] Warning, solution could not be verified
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -{y^{\prime }}^{3} x +2 {y^{\prime }}^{2} y-1=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 {\left (64 y^{3}+12 \sqrt {-96 y^{3}+81 x^{2}}\, x -108 x^{2}\right )^{\frac {1}{3}}}{6 x}+\frac {8 y^{2}}{3 x \left (64 y^{3}+12 \sqrt {-96 y^{3}+81 x^{2}}\, x -108 x^{2}\right )^{\frac {1}{3}}}+\frac {2 y}{3 x}, y^{\prime }=-\frac {\left (64 y^{3}+12 \sqrt {-96 y^{3}+81 x^{2}}\, x -108 x^{2}\right )^{\frac {1}{3}}}{12 x}-\frac {4 y^{2}}{3 x \left (64 y^{3}+12 \sqrt {-96 y^{3}+81 x^{2}}\, x -108 x^{2}\right )^{\frac {1}{3}}}+\frac {2 y}{3 x}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (64 y^{3}+12 \sqrt {-96 y^{3}+81 x^{2}}\, x -108 x^{2}\right )^{\frac {1}{3}}}{6 x}-\frac {8 y^{2}}{3 x \left (64 y^{3}+12 \sqrt {-96 y^{3}+81 x^{2}}\, x -108 x^{2}\right )^{\frac {1}{3}}}\right )}{2}, y^{\prime }=-\frac {\left (64 y^{3}+12 \sqrt {-96 y^{3}+81 x^{2}}\, x -108 x^{2}\right )^{\frac {1}{3}}}{12 x}-\frac {4 y^{2}}{3 x \left (64 y^{3}+12 \sqrt {-96 y^{3}+81 x^{2}}\, x -108 x^{2}\right )^{\frac {1}{3}}}+\frac {2 y}{3 x}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (64 y^{3}+12 \sqrt {-96 y^{3}+81 x^{2}}\, x -108 x^{2}\right )^{\frac {1}{3}}}{6 x}-\frac {8 y^{2}}{3 x \left (64 y^{3}+12 \sqrt {-96 y^{3}+81 x^{2}}\, x -108 x^{2}\right )^{\frac {1}{3}}}\right )}{2}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {\left (64 y^{3}+12 \sqrt {-96 y^{3}+81 x^{2}}\, x -108 x^{2}\right )^{\frac {1}{3}}}{6 x}+\frac {8 y^{2}}{3 x \left (64 y^{3}+12 \sqrt {-96 y^{3}+81 x^{2}}\, x -108 x^{2}\right )^{\frac {1}{3}}}+\frac {2 y}{3 x} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (64 y^{3}+12 \sqrt {-96 y^{3}+81 x^{2}}\, x -108 x^{2}\right )^{\frac {1}{3}}}{12 x}-\frac {4 y^{2}}{3 x \left (64 y^{3}+12 \sqrt {-96 y^{3}+81 x^{2}}\, x -108 x^{2}\right )^{\frac {1}{3}}}+\frac {2 y}{3 x}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (64 y^{3}+12 \sqrt {-96 y^{3}+81 x^{2}}\, x -108 x^{2}\right )^{\frac {1}{3}}}{6 x}-\frac {8 y^{2}}{3 x \left (64 y^{3}+12 \sqrt {-96 y^{3}+81 x^{2}}\, x -108 x^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (64 y^{3}+12 \sqrt {-96 y^{3}+81 x^{2}}\, x -108 x^{2}\right )^{\frac {1}{3}}}{12 x}-\frac {4 y^{2}}{3 x \left (64 y^{3}+12 \sqrt {-96 y^{3}+81 x^{2}}\, x -108 x^{2}\right )^{\frac {1}{3}}}+\frac {2 y}{3 x}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (64 y^{3}+12 \sqrt {-96 y^{3}+81 x^{2}}\, x -108 x^{2}\right )^{\frac {1}{3}}}{6 x}-\frac {8 y^{2}}{3 x \left (64 y^{3}+12 \sqrt {-96 y^{3}+81 x^{2}}\, x -108 x^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\mathit {workingODE} , \mathit {workingODE} , \mathit {workingODE}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: -> Solving 1st order ODE of high degree, 1st attempt trying 1st order WeierstrassP solution for high degree ODE trying 1st order WeierstrassPPrime solution for high degree ODE trying 1st order JacobiSN solution for high degree ODE trying 1st order ODE linearizable_by_differentiation trying differential order: 1; missing variables trying dAlembert <- dAlembert successful`
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 1441
dsolve(1/diff(y(x),x)^2+diff(y(x),x)*x=2*y(x),y(x), singsol=all)
\begin{align*} \frac {279936 \left (-\frac {x \left (-\frac {4 x^{2} y \left (x \right )^{2}}{9}-\frac {16 y \left (x \right )^{3} c_{1}}{3}+c_{1} x^{2}\right ) \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}}{108}-\frac {40 y \left (x \right )^{3} c_{1} x^{2}}{81}-\frac {x^{4} y \left (x \right )^{2}}{27}+\frac {c_{1} x^{4}}{12}+\frac {32 y \left (x \right )^{6} c_{1}}{81}+\frac {8 x^{2} y \left (x \right )^{5}}{243}\right ) \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}}+34992 \left (-\frac {16 y \left (x \right )^{3}}{27}+x^{2}-\frac {x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}}{9}\right ) \left (-\frac {16 x^{2} y \left (x \right )^{3}}{9}-\frac {64 c_{1} y \left (x \right )^{4}}{3}+x^{4}+\frac {32 x^{2} c_{1} y \left (x \right )}{3}\right ) \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}}+279936 y \left (x \right ) \left (-\frac {\left (-\frac {32 x^{2} y \left (x \right )^{3}}{27}-\frac {128 c_{1} y \left (x \right )^{4}}{9}+x^{4}+\frac {40 x^{2} c_{1} y \left (x \right )}{3}\right ) x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}}{9}-\frac {1792 y \left (x \right )^{4} c_{1} x^{2}}{81}+\frac {40 y \left (x \right ) c_{1} x^{4}}{3}-\frac {16 x^{4} y \left (x \right )^{3}}{9}+\frac {128 y \left (x \right )^{6} x^{2}}{243}+x^{6}+\frac {512 y \left (x \right )^{7} c_{1}}{81}\right )}{{\left (\left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}}+4 y \left (x \right ) \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}}+16 y \left (x \right )^{2}\right )}^{3} x \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}}} &= 0 \\ \frac {1119744 \left (\frac {\left (i-\frac {\sqrt {3}}{3}\right ) x \left (-\frac {16 x^{2} y \left (x \right )^{2}}{9}-\frac {16 y \left (x \right )^{3} c_{1}}{3}+c_{1} x^{2}\right ) \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}}{144}-\frac {\left (i \sqrt {3}-1\right ) \left (\frac {128 x^{2} y \left (x \right )^{5}}{81}+\frac {128 y \left (x \right )^{6} c_{1}}{27}-\frac {16 x^{4} y \left (x \right )^{2}}{9}-\frac {160 y \left (x \right )^{3} c_{1} x^{2}}{27}+c_{1} x^{4}\right )}{48}\right ) \left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}}-279936 \left (-\frac {x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}}{9}-\frac {16 y \left (x \right )^{3}}{27}+x^{2}\right ) \left (-\frac {16 x^{2} y \left (x \right )^{3}}{9}-\frac {16 c_{1} y \left (x \right )^{4}}{3}+x^{4}+\frac {8 x^{2} c_{1} y \left (x \right )}{3}\right ) \left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}}+1119744 y \left (x \right ) \left (-\frac {\left (i+\frac {\sqrt {3}}{3}\right ) \left (-\frac {32 x^{2} y \left (x \right )^{3}}{27}-\frac {32 c_{1} y \left (x \right )^{4}}{9}+x^{4}+\frac {10 x^{2} c_{1} y \left (x \right )}{3}\right ) x \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}}{3}+\left (\frac {128 y \left (x \right )^{6} x^{2}}{243}+\frac {128 y \left (x \right )^{7} c_{1}}{81}-\frac {16 x^{4} y \left (x \right )^{3}}{9}-\frac {448 y \left (x \right )^{4} c_{1} x^{2}}{81}+x^{6}+\frac {10 y \left (x \right ) c_{1} x^{4}}{3}\right ) \left (1+i \sqrt {3}\right )\right )}{\left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}} \left (16 i \sqrt {3}\, y \left (x \right )^{2}-i \left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}} \sqrt {3}+16 y \left (x \right )^{2}-8 y \left (x \right ) \left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}}+\left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}}\right )^{3} x} &= 0 \\ \frac {1119744 \left (\frac {\left (i+\frac {\sqrt {3}}{3}\right ) x \left (-\frac {16 x^{2} y \left (x \right )^{2}}{9}-\frac {16 y \left (x \right )^{3} c_{1}}{3}+c_{1} x^{2}\right ) \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}}{144}-\frac {\left (\frac {128 x^{2} y \left (x \right )^{5}}{81}+\frac {128 y \left (x \right )^{6} c_{1}}{27}-\frac {16 x^{4} y \left (x \right )^{2}}{9}-\frac {160 y \left (x \right )^{3} c_{1} x^{2}}{27}+c_{1} x^{4}\right ) \left (1+i \sqrt {3}\right )}{48}\right ) \left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}}+279936 \left (-\frac {x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}}{9}-\frac {16 y \left (x \right )^{3}}{27}+x^{2}\right ) \left (-\frac {16 x^{2} y \left (x \right )^{3}}{9}-\frac {16 c_{1} y \left (x \right )^{4}}{3}+x^{4}+\frac {8 x^{2} c_{1} y \left (x \right )}{3}\right ) \left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}}+1119744 \left (-\frac {\left (-\frac {32 x^{2} y \left (x \right )^{3}}{27}-\frac {32 c_{1} y \left (x \right )^{4}}{9}+x^{4}+\frac {10 x^{2} c_{1} y \left (x \right )}{3}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) x \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}}{3}+\left (\frac {128 y \left (x \right )^{6} x^{2}}{243}+\frac {128 y \left (x \right )^{7} c_{1}}{81}-\frac {16 x^{4} y \left (x \right )^{3}}{9}-\frac {448 y \left (x \right )^{4} c_{1} x^{2}}{81}+x^{6}+\frac {10 y \left (x \right ) c_{1} x^{4}}{3}\right ) \left (i \sqrt {3}-1\right )\right ) y \left (x \right )}{\left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}} \left (16 i \sqrt {3}\, y \left (x \right )^{2}-i \left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}} \sqrt {3}-16 y \left (x \right )^{2}+8 y \left (x \right ) \left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}}-\left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}}\right )^{3} x} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 149.881 (sec). Leaf size: 10773
DSolve[1/(y'[x]^2)+y'[x]*x==2*y[x],y[x],x,IncludeSingularSolutions -> True]
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