Internal problem ID [8813]
Internal file name [OUTPUT/7748_Sunday_June_05_2022_11_56_40_PM_862407/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 479.
ODE order: 1.
ODE degree: 2.
The type(s) of ODE detected by this program : "dAlembert"
Maple gives the following as the ode type
[_rational, _dAlembert]
\[ \boxed {\left (b_{2} y+a_{2} x +c_{2} \right ) {y^{\prime }}^{2}+\left (a_{1} x +b_{1} y+c_{1} \right ) y^{\prime }+b_{0} y=-a_{0} x -c_{0}} \]
Let \(p=y^{\prime }\) the ode becomes \begin {align*} \left (a_{2} x +b_{2} y +c_{2} \right ) p^{2}+\left (a_{1} x +b_{1} y +c_{1} \right ) p +b_{0} y = -a_{0} x -c_{0} \end {align*}
Solving for \(y\) from the above results in \begin {align*} y &= -\frac {\left (a_{2} p^{2}+a_{1} p +a_{0} \right ) x}{b_{2} p^{2}+b_{1} p +b_{0}}-\frac {c_{2} p^{2}+c_{1} p +c_{0}}{b_{2} p^{2}+b_{1} p +b_{0}}\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 {-a_{2} p^{2}-a_{1} p -a_{0}}{b_{2} p^{2}+b_{1} p +b_{0}}\\ g &= \frac {-c_{2} p^{2}-c_{1} p -c_{0}}{b_{2} p^{2}+b_{1} p +b_{0}} \end {align*}
Hence (2) becomes \begin {align*} p -\frac {-a_{2} p^{2}-a_{1} p -a_{0}}{b_{2} p^{2}+b_{1} p +b_{0}} = \left (x \left (\frac {-2 a_{2} p -a_{1}}{b_{2} p^{2}+b_{1} p +b_{0}}-\frac {\left (-a_{2} p^{2}-a_{1} p -a_{0} \right ) \left (2 b_{2} p +b_{1} \right )}{\left (b_{2} p^{2}+b_{1} p +b_{0} \right )^{2}}\right )+\frac {-2 c_{2} p -c_{1}}{b_{2} p^{2}+b_{1} p +b_{0}}-\frac {\left (-c_{2} p^{2}-c_{1} p -c_{0} \right ) \left (2 b_{2} p +b_{1} \right )}{\left (b_{2} p^{2}+b_{1} p +b_{0} \right )^{2}}\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*} p -\frac {-a_{2} p^{2}-a_{1} p -a_{0}}{b_{2} p^{2}+b_{1} p +b_{0}} = 0 \end {align*}
Solving for \(p\) from the above gives \begin {align*} p&=\frac {\left (12 \sqrt {3}\, \sqrt {27 a_{0}^{2} b_{2}^{2}-18 a_{0} a_{1} a_{2} b_{2} -18 a_{0} a_{1} b_{1} b_{2} +4 a_{0} a_{2}^{3}+12 a_{0} a_{2}^{2} b_{1} -18 a_{0} a_{2} b_{0} b_{2} +12 a_{0} a_{2} b_{1}^{2}-18 a_{0} b_{0} b_{1} b_{2} +4 a_{0} b_{1}^{3}+4 a_{1}^{3} b_{2} -a_{1}^{2} a_{2}^{2}-2 a_{1}^{2} a_{2} b_{1} +12 a_{1}^{2} b_{0} b_{2} -a_{1}^{2} b_{1}^{2}-2 a_{1} a_{2}^{2} b_{0} -4 a_{1} a_{2} b_{0} b_{1} +12 a_{1} b_{0}^{2} b_{2} -2 a_{1} b_{0} b_{1}^{2}-a_{2}^{2} b_{0}^{2}-2 a_{2} b_{0}^{2} b_{1} +4 b_{0}^{3} b_{2} -b_{0}^{2} b_{1}^{2}}\, b_{2} -108 a_{0} b_{2}^{2}+36 a_{1} a_{2} b_{2} +36 a_{1} b_{1} b_{2} -8 a_{2}^{3}-24 a_{2}^{2} b_{1} +36 a_{2} b_{0} b_{2} -24 a_{2} b_{1}^{2}+36 b_{0} b_{1} b_{2} -8 b_{1}^{3}\right )^{\frac {1}{3}}}{6 b_{2}}-\frac {2 \left (3 b_{2} a_{1} -a_{2}^{2}-2 a_{2} b_{1} +3 b_{2} b_{0} -b_{1}^{2}\right )}{3 b_{2} \left (12 \sqrt {3}\, \sqrt {27 a_{0}^{2} b_{2}^{2}-18 a_{0} a_{1} a_{2} b_{2} -18 a_{0} a_{1} b_{1} b_{2} +4 a_{0} a_{2}^{3}+12 a_{0} a_{2}^{2} b_{1} -18 a_{0} a_{2} b_{0} b_{2} +12 a_{0} a_{2} b_{1}^{2}-18 a_{0} b_{0} b_{1} b_{2} +4 a_{0} b_{1}^{3}+4 a_{1}^{3} b_{2} -a_{1}^{2} a_{2}^{2}-2 a_{1}^{2} a_{2} b_{1} +12 a_{1}^{2} b_{0} b_{2} -a_{1}^{2} b_{1}^{2}-2 a_{1} a_{2}^{2} b_{0} -4 a_{1} a_{2} b_{0} b_{1} +12 a_{1} b_{0}^{2} b_{2} -2 a_{1} b_{0} b_{1}^{2}-a_{2}^{2} b_{0}^{2}-2 a_{2} b_{0}^{2} b_{1} +4 b_{0}^{3} b_{2} -b_{0}^{2} b_{1}^{2}}\, b_{2} -108 a_{0} b_{2}^{2}+36 a_{1} a_{2} b_{2} +36 a_{1} b_{1} b_{2} -8 a_{2}^{3}-24 a_{2}^{2} b_{1} +36 a_{2} b_{0} b_{2} -24 a_{2} b_{1}^{2}+36 b_{0} b_{1} b_{2} -8 b_{1}^{3}\right )^{\frac {1}{3}}}-\frac {a_{2} +b_{1}}{3 b_{2}}\\ p&=\text {Expression too large to display}\\ p&=\text {Expression too large to display} \end {align*}
Substituting these in (1A) gives \begin {align*} y&=\text {Expression too large to display}\\ y&=\text {Expression too large to display}\\ y&=\text {Expression too large to display} \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 )-\frac {-a_{2} p \left (x \right )^{2}-a_{1} p \left (x \right )-a_{0}}{b_{2} p \left (x \right )^{2}+b_{1} p \left (x \right )+b_{0}}}{x \left (\frac {-2 a_{2} p \left (x \right )-a_{1}}{b_{2} p \left (x \right )^{2}+b_{1} p \left (x \right )+b_{0}}-\frac {\left (-a_{2} p \left (x \right )^{2}-a_{1} p \left (x \right )-a_{0} \right ) \left (2 b_{2} p \left (x \right )+b_{1} \right )}{\left (b_{2} p \left (x \right )^{2}+b_{1} p \left (x \right )+b_{0} \right )^{2}}\right )+\frac {-2 c_{2} p \left (x \right )-c_{1}}{b_{2} p \left (x \right )^{2}+b_{1} p \left (x \right )+b_{0}}-\frac {\left (-c_{2} p \left (x \right )^{2}-c_{1} p \left (x \right )-c_{0} \right ) \left (2 b_{2} p \left (x \right )+b_{1} \right )}{\left (b_{2} p \left (x \right )^{2}+b_{1} p \left (x \right )+b_{0} \right )^{2}}}\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 ) \left (\frac {-2 a_{2} p -a_{1}}{b_{2} p^{2}+b_{1} p +b_{0}}-\frac {\left (-a_{2} p^{2}-a_{1} p -a_{0} \right ) \left (2 b_{2} p +b_{1} \right )}{\left (b_{2} p^{2}+b_{1} p +b_{0} \right )^{2}}\right )+\frac {-2 c_{2} p -c_{1}}{b_{2} p^{2}+b_{1} p +b_{0}}-\frac {\left (-c_{2} p^{2}-c_{1} p -c_{0} \right ) \left (2 b_{2} p +b_{1} \right )}{\left (b_{2} p^{2}+b_{1} p +b_{0} \right )^{2}}}{p -\frac {-a_{2} p^{2}-a_{1} p -a_{0}}{b_{2} p^{2}+b_{1} p +b_{0}}}\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 {a_{1} b_{2} p^{2}-a_{2} b_{1} p^{2}+2 a_{0} b_{2} p -2 a_{2} b_{0} p +b_{1} a_{0} -b_{0} a_{1}}{\left (b_{2} p^{3}+a_{2} p^{2}+b_{1} p^{2}+a_{1} p +b_{0} p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )}\\ q(p) &=\frac {-b_{1} c_{2} p^{2}+b_{2} c_{1} p^{2}-2 b_{0} c_{2} p +2 b_{2} c_{0} p -b_{0} c_{1} +b_{1} c_{0}}{\left (b_{2} p^{3}+a_{2} p^{2}+b_{1} p^{2}+a_{1} p +b_{0} p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )} \end {align*}
Hence the ode is \begin {align*} \frac {d}{d p}x \left (p \right )-\frac {\left (a_{1} b_{2} p^{2}-a_{2} b_{1} p^{2}+2 a_{0} b_{2} p -2 a_{2} b_{0} p +b_{1} a_{0} -b_{0} a_{1} \right ) x \left (p \right )}{\left (b_{2} p^{3}+a_{2} p^{2}+b_{1} p^{2}+a_{1} p +b_{0} p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )} = \frac {-b_{1} c_{2} p^{2}+b_{2} c_{1} p^{2}-2 b_{0} c_{2} p +2 b_{2} c_{0} p -b_{0} c_{1} +b_{1} c_{0}}{\left (b_{2} p^{3}+a_{2} p^{2}+b_{1} p^{2}+a_{1} p +b_{0} p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )} \end {align*}
The integrating factor \(\mu \) is \[ \mu = {\mathrm e}^{\int -\frac {a_{1} b_{2} p^{2}-a_{2} b_{1} p^{2}+2 a_{0} b_{2} p -2 a_{2} b_{0} p +b_{1} a_{0} -b_{0} a_{1}}{\left (b_{2} p^{3}+a_{2} p^{2}+b_{1} p^{2}+a_{1} p +b_{0} p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )}d p} \] The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}p}}\left ( \mu x\right ) &= \left (\mu \right ) \left (\frac {-b_{1} c_{2} p^{2}+b_{2} c_{1} p^{2}-2 b_{0} c_{2} p +2 b_{2} c_{0} p -b_{0} c_{1} +b_{1} c_{0}}{\left (b_{2} p^{3}+a_{2} p^{2}+b_{1} p^{2}+a_{1} p +b_{0} p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}p}} \left ({\mathrm e}^{\int -\frac {a_{1} b_{2} p^{2}-a_{2} b_{1} p^{2}+2 a_{0} b_{2} p -2 a_{2} b_{0} p +b_{1} a_{0} -b_{0} a_{1}}{\left (b_{2} p^{3}+a_{2} p^{2}+b_{1} p^{2}+a_{1} p +b_{0} p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )}d p} x\right ) &= \left ({\mathrm e}^{\int -\frac {a_{1} b_{2} p^{2}-a_{2} b_{1} p^{2}+2 a_{0} b_{2} p -2 a_{2} b_{0} p +b_{1} a_{0} -b_{0} a_{1}}{\left (b_{2} p^{3}+a_{2} p^{2}+b_{1} p^{2}+a_{1} p +b_{0} p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )}d p}\right ) \left (\frac {-b_{1} c_{2} p^{2}+b_{2} c_{1} p^{2}-2 b_{0} c_{2} p +2 b_{2} c_{0} p -b_{0} c_{1} +b_{1} c_{0}}{\left (b_{2} p^{3}+a_{2} p^{2}+b_{1} p^{2}+a_{1} p +b_{0} p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )}\right )\\ \mathrm {d} \left ({\mathrm e}^{\int -\frac {a_{1} b_{2} p^{2}-a_{2} b_{1} p^{2}+2 a_{0} b_{2} p -2 a_{2} b_{0} p +b_{1} a_{0} -b_{0} a_{1}}{\left (b_{2} p^{3}+a_{2} p^{2}+b_{1} p^{2}+a_{1} p +b_{0} p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )}d p} x\right ) &= \left (\frac {\left (\left (-b_{1} c_{2} +b_{2} c_{1} \right ) p^{2}+\left (-2 b_{0} c_{2} +2 b_{2} c_{0} \right ) p +b_{1} c_{0} -b_{0} c_{1} \right ) {\mathrm e}^{-\left (\int \frac {\left (b_{2} a_{1} -a_{2} b_{1} \right ) p^{2}+\left (2 a_{0} b_{2} -2 a_{2} b_{0} \right ) p +b_{1} a_{0} -b_{0} a_{1}}{\left (b_{2} p^{3}+\left (a_{2} +b_{1} \right ) p^{2}+\left (a_{1} +b_{0} \right ) p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )}d p \right )}}{\left (b_{2} p^{3}+\left (a_{2} +b_{1} \right ) p^{2}+\left (a_{1} +b_{0} \right ) p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )}\right )\, \mathrm {d} p \end {align*}
Integrating gives \begin {align*} {\mathrm e}^{\int -\frac {a_{1} b_{2} p^{2}-a_{2} b_{1} p^{2}+2 a_{0} b_{2} p -2 a_{2} b_{0} p +b_{1} a_{0} -b_{0} a_{1}}{\left (b_{2} p^{3}+a_{2} p^{2}+b_{1} p^{2}+a_{1} p +b_{0} p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )}d p} x &= \int {\frac {\left (\left (-b_{1} c_{2} +b_{2} c_{1} \right ) p^{2}+\left (-2 b_{0} c_{2} +2 b_{2} c_{0} \right ) p +b_{1} c_{0} -b_{0} c_{1} \right ) {\mathrm e}^{-\left (\int \frac {\left (b_{2} a_{1} -a_{2} b_{1} \right ) p^{2}+\left (2 a_{0} b_{2} -2 a_{2} b_{0} \right ) p +b_{1} a_{0} -b_{0} a_{1}}{\left (b_{2} p^{3}+\left (a_{2} +b_{1} \right ) p^{2}+\left (a_{1} +b_{0} \right ) p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )}d p \right )}}{\left (b_{2} p^{3}+\left (a_{2} +b_{1} \right ) p^{2}+\left (a_{1} +b_{0} \right ) p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )}\,\mathrm {d} p}\\ {\mathrm e}^{\int -\frac {a_{1} b_{2} p^{2}-a_{2} b_{1} p^{2}+2 a_{0} b_{2} p -2 a_{2} b_{0} p +b_{1} a_{0} -b_{0} a_{1}}{\left (b_{2} p^{3}+a_{2} p^{2}+b_{1} p^{2}+a_{1} p +b_{0} p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )}d p} x &= \int \frac {\left (\left (-b_{1} c_{2} +b_{2} c_{1} \right ) p^{2}+\left (-2 b_{0} c_{2} +2 b_{2} c_{0} \right ) p +b_{1} c_{0} -b_{0} c_{1} \right ) {\mathrm e}^{-\left (\int \frac {\left (b_{2} a_{1} -a_{2} b_{1} \right ) p^{2}+\left (2 a_{0} b_{2} -2 a_{2} b_{0} \right ) p +b_{1} a_{0} -b_{0} a_{1}}{\left (b_{2} p^{3}+\left (a_{2} +b_{1} \right ) p^{2}+\left (a_{1} +b_{0} \right ) p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )}d p \right )}}{\left (b_{2} p^{3}+\left (a_{2} +b_{1} \right ) p^{2}+\left (a_{1} +b_{0} \right ) p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )}d p + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu ={\mathrm e}^{\int -\frac {a_{1} b_{2} p^{2}-a_{2} b_{1} p^{2}+2 a_{0} b_{2} p -2 a_{2} b_{0} p +b_{1} a_{0} -b_{0} a_{1}}{\left (b_{2} p^{3}+a_{2} p^{2}+b_{1} p^{2}+a_{1} p +b_{0} p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )}d p}\) results in \begin {align*} x \left (p \right ) &= {\mathrm e}^{\int \frac {\left (b_{2} a_{1} -a_{2} b_{1} \right ) p^{2}+\left (2 a_{0} b_{2} -2 a_{2} b_{0} \right ) p +b_{1} a_{0} -b_{0} a_{1}}{\left (b_{2} p^{3}+\left (a_{2} +b_{1} \right ) p^{2}+\left (a_{1} +b_{0} \right ) p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )}d p} \left (\int \frac {\left (\left (-b_{1} c_{2} +b_{2} c_{1} \right ) p^{2}+\left (-2 b_{0} c_{2} +2 b_{2} c_{0} \right ) p +b_{1} c_{0} -b_{0} c_{1} \right ) {\mathrm e}^{-\left (\int \frac {\left (b_{2} a_{1} -a_{2} b_{1} \right ) p^{2}+\left (2 a_{0} b_{2} -2 a_{2} b_{0} \right ) p +b_{1} a_{0} -b_{0} a_{1}}{\left (b_{2} p^{3}+\left (a_{2} +b_{1} \right ) p^{2}+\left (a_{1} +b_{0} \right ) p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )}d p \right )}}{\left (b_{2} p^{3}+\left (a_{2} +b_{1} \right ) p^{2}+\left (a_{1} +b_{0} \right ) p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )}d p \right )+c_{1} {\mathrm e}^{\int \frac {\left (b_{2} a_{1} -a_{2} b_{1} \right ) p^{2}+\left (2 a_{0} b_{2} -2 a_{2} b_{0} \right ) p +b_{1} a_{0} -b_{0} a_{1}}{\left (b_{2} p^{3}+\left (a_{2} +b_{1} \right ) p^{2}+\left (a_{1} +b_{0} \right ) p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )}d p} \end {align*}
which simplifies to \begin {align*} x \left (p \right ) &= {\mathrm e}^{\int \frac {\left (b_{2} a_{1} -a_{2} b_{1} \right ) p^{2}+\left (2 a_{0} b_{2} -2 a_{2} b_{0} \right ) p +b_{1} a_{0} -b_{0} a_{1}}{\left (b_{2} p^{3}+\left (a_{2} +b_{1} \right ) p^{2}+\left (a_{1} +b_{0} \right ) p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )}d p} \left (-\left (\int -\frac {\left (-b_{1} c_{2} p^{2}+b_{2} c_{1} p^{2}-2 b_{0} c_{2} p +2 b_{2} c_{0} p -b_{0} c_{1} +b_{1} c_{0} \right ) {\mathrm e}^{-\left (\int \frac {\left (b_{2} a_{1} -a_{2} b_{1} \right ) p^{2}+\left (2 a_{0} b_{2} -2 a_{2} b_{0} \right ) p +b_{1} a_{0} -b_{0} a_{1}}{\left (b_{2} p^{3}+\left (a_{2} +b_{1} \right ) p^{2}+\left (a_{1} +b_{0} \right ) p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )}d p \right )}}{\left (b_{2} p^{3}+\left (a_{2} +b_{1} \right ) p^{2}+\left (a_{1} +b_{0} \right ) p +a_{0} \right ) \left (b_{2} p^{2}+b_{1} p +b_{0} \right )}d p \right )+c_{1} \right ) \end {align*}
Since the solution \(x \left (p \right )\) has unresolved integral, unable to continue.
Summary
The solution(s) found are the following \begin{align*} \tag{1} \text {Expression too large to display} \\ \tag{2} \text {Expression too large to display} \\ \tag{3} \text {Expression too large to display} \\ \end{align*}
Verification of solutions
\[ \text {Expression too large to display} \] Warning, solution could not be verified
\[ \text {Expression too large to display} \] Warning, solution could not be verified
\[ \text {Expression too large to display} \] Warning, solution could not be verified
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (b_{2} y+a_{2} x +c_{2} \right ) {y^{\prime }}^{2}+\left (a_{1} x +b_{1} y+c_{1} \right ) y^{\prime }+b_{0} y=-a_{0} x -c_{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 {-a_{1} x -b_{1} y-c_{1} +\sqrt {-4 y^{2} b_{0} b_{2} +y^{2} b_{1}^{2}-4 y a_{0} b_{2} x +2 y a_{1} b_{1} x -4 y a_{2} b_{0} x -4 a_{0} a_{2} x^{2}+a_{1}^{2} x^{2}+2 y c_{1} b_{1} -4 y c_{2} b_{0} -4 y b_{2} c_{0} +2 c_{1} a_{1} x -4 c_{2} a_{0} x -4 a_{2} c_{0} x +c_{1}^{2}-4 c_{2} c_{0}}}{2 \left (b_{2} y+a_{2} x +c_{2} \right )}, y^{\prime }=-\frac {b_{1} y+a_{1} x +\sqrt {-4 y^{2} b_{0} b_{2} +y^{2} b_{1}^{2}-4 y a_{0} b_{2} x +2 y a_{1} b_{1} x -4 y a_{2} b_{0} x -4 a_{0} a_{2} x^{2}+a_{1}^{2} x^{2}+2 y c_{1} b_{1} -4 y c_{2} b_{0} -4 y b_{2} c_{0} +2 c_{1} a_{1} x -4 c_{2} a_{0} x -4 a_{2} c_{0} x +c_{1}^{2}-4 c_{2} c_{0}}+c_{1}}{2 \left (b_{2} y+a_{2} x +c_{2} \right )}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {-a_{1} x -b_{1} y-c_{1} +\sqrt {-4 y^{2} b_{0} b_{2} +y^{2} b_{1}^{2}-4 y a_{0} b_{2} x +2 y a_{1} b_{1} x -4 y a_{2} b_{0} x -4 a_{0} a_{2} x^{2}+a_{1}^{2} x^{2}+2 y c_{1} b_{1} -4 y c_{2} b_{0} -4 y b_{2} c_{0} +2 c_{1} a_{1} x -4 c_{2} a_{0} x -4 a_{2} c_{0} x +c_{1}^{2}-4 c_{2} c_{0}}}{2 \left (b_{2} y+a_{2} x +c_{2} \right )} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {b_{1} y+a_{1} x +\sqrt {-4 y^{2} b_{0} b_{2} +y^{2} b_{1}^{2}-4 y a_{0} b_{2} x +2 y a_{1} b_{1} x -4 y a_{2} b_{0} x -4 a_{0} a_{2} x^{2}+a_{1}^{2} x^{2}+2 y c_{1} b_{1} -4 y c_{2} b_{0} -4 y b_{2} c_{0} +2 c_{1} a_{1} x -4 c_{2} a_{0} x -4 a_{2} c_{0} x +c_{1}^{2}-4 c_{2} c_{0}}+c_{1}}{2 \left (b_{2} y+a_{2} x +c_{2} \right )} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\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: 1.234 (sec). Leaf size: 875
dsolve((b__2*y(x)+a__2*x+c__2)*diff(y(x),x)^2+(a__1*x+b__1*y(x)+c__1)*diff(y(x),x)+a__0*x+b__0*y(x)+c__0 = 0,y(x), singsol=all)
\begin{align*} x -{\mathrm e}^{\int _{}^{\frac {-a_{1} x -b_{1} y \left (x \right )-c_{1} -\sqrt {\left (-4 b_{0} b_{2} +b_{1}^{2}\right ) y \left (x \right )^{2}+\left (\left (-4 a_{0} b_{2} +2 a_{1} b_{1} -4 a_{2} b_{0} \right ) x -4 b_{2} c_{0} +2 c_{1} b_{1} -4 c_{2} b_{0} \right ) y \left (x \right )+\left (-4 a_{0} a_{2} +a_{1}^{2}\right ) x^{2}+\left (-4 a_{0} c_{2} +2 a_{1} c_{1} -4 c_{0} a_{2} \right ) x -4 c_{2} c_{0} +c_{1}^{2}}}{2 b_{2} y \left (x \right )+2 c_{2} +2 a_{2} x}}\frac {\left (a_{1} b_{2} -a_{2} b_{1} \right ) \textit {\_a}^{2}+\left (2 a_{0} b_{2} -2 a_{2} b_{0} \right ) \textit {\_a} +a_{0} b_{1} -b_{0} a_{1}}{\left (\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0} \right ) \left (\textit {\_a}^{3} b_{2} +\left (a_{2} +b_{1} \right ) \textit {\_a}^{2}+\left (a_{1} +b_{0} \right ) \textit {\_a} +a_{0} \right )}d \textit {\_a}} \left (\int _{}^{\frac {-a_{1} x -b_{1} y \left (x \right )-c_{1} -\sqrt {\left (-4 b_{0} b_{2} +b_{1}^{2}\right ) y \left (x \right )^{2}+\left (\left (-4 a_{0} b_{2} +2 a_{1} b_{1} -4 a_{2} b_{0} \right ) x -4 b_{2} c_{0} +2 c_{1} b_{1} -4 c_{2} b_{0} \right ) y \left (x \right )+\left (-4 a_{0} a_{2} +a_{1}^{2}\right ) x^{2}+\left (-4 a_{0} c_{2} +2 a_{1} c_{1} -4 c_{0} a_{2} \right ) x -4 c_{2} c_{0} +c_{1}^{2}}}{2 b_{2} y \left (x \right )+2 c_{2} +2 a_{2} x}}\frac {\left (-\textit {\_a}^{2} b_{1} c_{2} +\textit {\_a}^{2} b_{2} c_{1} -2 \textit {\_a} b_{0} c_{2} +2 \textit {\_a} b_{2} c_{0} -b_{0} c_{1} +c_{0} b_{1} \right ) {\mathrm e}^{-\left (\int \frac {\left (a_{1} b_{2} -a_{2} b_{1} \right ) \textit {\_a}^{2}+\left (2 a_{0} b_{2} -2 a_{2} b_{0} \right ) \textit {\_a} +a_{0} b_{1} -b_{0} a_{1}}{\left (\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0} \right ) \left (\textit {\_a}^{3} b_{2} +\left (a_{2} +b_{1} \right ) \textit {\_a}^{2}+\left (a_{1} +b_{0} \right ) \textit {\_a} +a_{0} \right )}d \textit {\_a} \right )}}{\left (\textit {\_a}^{3} b_{2} +\left (a_{2} +b_{1} \right ) \textit {\_a}^{2}+\left (a_{1} +b_{0} \right ) \textit {\_a} +a_{0} \right ) \left (\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0} \right )}d \textit {\_a} +c_{3} \right ) &= 0 \\ x -{\mathrm e}^{\int _{}^{\frac {-a_{1} x -b_{1} y \left (x \right )-c_{1} +\sqrt {\left (-4 b_{0} b_{2} +b_{1}^{2}\right ) y \left (x \right )^{2}+\left (\left (-4 a_{0} b_{2} +2 a_{1} b_{1} -4 a_{2} b_{0} \right ) x -4 b_{2} c_{0} +2 c_{1} b_{1} -4 c_{2} b_{0} \right ) y \left (x \right )+\left (-4 a_{0} a_{2} +a_{1}^{2}\right ) x^{2}+\left (-4 a_{0} c_{2} +2 a_{1} c_{1} -4 c_{0} a_{2} \right ) x -4 c_{2} c_{0} +c_{1}^{2}}}{2 b_{2} y \left (x \right )+2 c_{2} +2 a_{2} x}}\frac {\left (a_{1} b_{2} -a_{2} b_{1} \right ) \textit {\_a}^{2}+\left (2 a_{0} b_{2} -2 a_{2} b_{0} \right ) \textit {\_a} +a_{0} b_{1} -b_{0} a_{1}}{\left (\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0} \right ) \left (\textit {\_a}^{3} b_{2} +\left (a_{2} +b_{1} \right ) \textit {\_a}^{2}+\left (a_{1} +b_{0} \right ) \textit {\_a} +a_{0} \right )}d \textit {\_a}} \left (\int _{}^{\frac {-a_{1} x -b_{1} y \left (x \right )-c_{1} +\sqrt {\left (-4 b_{0} b_{2} +b_{1}^{2}\right ) y \left (x \right )^{2}+\left (\left (-4 a_{0} b_{2} +2 a_{1} b_{1} -4 a_{2} b_{0} \right ) x -4 b_{2} c_{0} +2 c_{1} b_{1} -4 c_{2} b_{0} \right ) y \left (x \right )+\left (-4 a_{0} a_{2} +a_{1}^{2}\right ) x^{2}+\left (-4 a_{0} c_{2} +2 a_{1} c_{1} -4 c_{0} a_{2} \right ) x -4 c_{2} c_{0} +c_{1}^{2}}}{2 b_{2} y \left (x \right )+2 c_{2} +2 a_{2} x}}\frac {\left (-\textit {\_a}^{2} b_{1} c_{2} +\textit {\_a}^{2} b_{2} c_{1} -2 \textit {\_a} b_{0} c_{2} +2 \textit {\_a} b_{2} c_{0} -b_{0} c_{1} +c_{0} b_{1} \right ) {\mathrm e}^{-\left (\int \frac {\left (a_{1} b_{2} -a_{2} b_{1} \right ) \textit {\_a}^{2}+\left (2 a_{0} b_{2} -2 a_{2} b_{0} \right ) \textit {\_a} +a_{0} b_{1} -b_{0} a_{1}}{\left (\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0} \right ) \left (\textit {\_a}^{3} b_{2} +\left (a_{2} +b_{1} \right ) \textit {\_a}^{2}+\left (a_{1} +b_{0} \right ) \textit {\_a} +a_{0} \right )}d \textit {\_a} \right )}}{\left (\textit {\_a}^{3} b_{2} +\left (a_{2} +b_{1} \right ) \textit {\_a}^{2}+\left (a_{1} +b_{0} \right ) \textit {\_a} +a_{0} \right ) \left (\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0} \right )}d \textit {\_a} +c_{3} \right ) &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.811 (sec). Leaf size: 576
DSolve[c0 + a0*x + b0*y[x] + (c1 + a1*x + b1*y[x])*y'[x] + (c2 + a2*x + b2*y[x])*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{x=-\frac {-(K[2] (\text {b2} K[2]+\text {b1})+\text {b0}) \exp \left (\text {RootSum}\left [\text {$\#$1}^3 \text {b2}+\text {$\#$1}^2 \text {a2}+\text {$\#$1}^2 \text {b1}+\text {$\#$1} \text {a1}+\text {$\#$1} \text {b0}+\text {a0}\&,\frac {\text {$\#$1}^2 \text {b2} \log (K[2]-\text {$\#$1})+\text {b0} \log (K[2]-\text {$\#$1})+\text {$\#$1} \text {b1} \log (K[2]-\text {$\#$1})}{3 \text {$\#$1}^2 \text {b2}+2 \text {$\#$1} \text {a2}+2 \text {$\#$1} \text {b1}+\text {a1}+\text {b0}}\&\right ]\right ) \left (\int _1^{K[2]}\frac {\exp \left (-\text {RootSum}\left [\text {b2} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {b1} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {b0} \text {$\#$1}+\text {a0}\&,\frac {\text {b2} \log (K[1]-\text {$\#$1}) \text {$\#$1}^2+\text {b1} \log (K[1]-\text {$\#$1}) \text {$\#$1}+\text {b0} \log (K[1]-\text {$\#$1})}{3 \text {b2} \text {$\#$1}^2+2 \text {a2} \text {$\#$1}+2 \text {b1} \text {$\#$1}+\text {a1}+\text {b0}}\&\right ]\right ) (-\text {c0}-K[1] (\text {c1}+\text {c2} K[1]))}{\text {a0}+K[1] (\text {a1}+\text {b0}+K[1] (\text {a2}+\text {b1}+\text {b2} K[1]))}dK[1]+c_1\right )+\text {c1} K[2]+\text {c2} K[2]^2+\text {c0}}{K[2] (K[2] (\text {b2} K[2]+\text {a2}+\text {b1})+\text {a1}+\text {b0})+\text {a0}},y(x)=-\frac {K[2] (K[2] (\text {c2} K[2]+\text {c1})+\text {c0})+(K[2] (\text {a2} K[2]+\text {a1})+\text {a0}) \exp \left (\text {RootSum}\left [\text {$\#$1}^3 \text {b2}+\text {$\#$1}^2 \text {a2}+\text {$\#$1}^2 \text {b1}+\text {$\#$1} \text {a1}+\text {$\#$1} \text {b0}+\text {a0}\&,\frac {\text {$\#$1}^2 \text {b2} \log (K[2]-\text {$\#$1})+\text {b0} \log (K[2]-\text {$\#$1})+\text {$\#$1} \text {b1} \log (K[2]-\text {$\#$1})}{3 \text {$\#$1}^2 \text {b2}+2 \text {$\#$1} \text {a2}+2 \text {$\#$1} \text {b1}+\text {a1}+\text {b0}}\&\right ]\right ) \left (\int _1^{K[2]}\frac {\exp \left (-\text {RootSum}\left [\text {b2} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {b1} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {b0} \text {$\#$1}+\text {a0}\&,\frac {\text {b2} \log (K[1]-\text {$\#$1}) \text {$\#$1}^2+\text {b1} \log (K[1]-\text {$\#$1}) \text {$\#$1}+\text {b0} \log (K[1]-\text {$\#$1})}{3 \text {b2} \text {$\#$1}^2+2 \text {a2} \text {$\#$1}+2 \text {b1} \text {$\#$1}+\text {a1}+\text {b0}}\&\right ]\right ) (-\text {c0}-K[1] (\text {c1}+\text {c2} K[1]))}{\text {a0}+K[1] (\text {a1}+\text {b0}+K[1] (\text {a2}+\text {b1}+\text {b2} K[1]))}dK[1]+c_1\right )}{K[2] (K[2] (\text {b2} K[2]+\text {a2}+\text {b1})+\text {a1}+\text {b0})+\text {a0}}\right \},\{y(x),K[2]\}\right ] \]