Internal problem ID [8765]
Internal file name [OUTPUT/7699_Sunday_June_05_2022_11_42_50_PM_73250892/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 430.
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]
\[ \boxed {\left (\operatorname {a2} x +\operatorname {c2} \right ) {y^{\prime }}^{2}+\left (\operatorname {a1} x +\operatorname {b1} y+\operatorname {c1} \right ) y^{\prime }+\operatorname {b0} y=-\operatorname {a0} x -\operatorname {c0}} \]
Let \(p=y^{\prime }\) the ode becomes \begin {align*} \left (\operatorname {a2} x +\operatorname {c2} \right ) p^{2}+\left (\operatorname {a1} x +\operatorname {b1} y +\operatorname {c1} \right ) p +y \operatorname {b0} = -\operatorname {a0} x -\operatorname {c0} \end {align*}
Solving for \(y\) from the above results in \begin {align*} y &= -\frac {\left (\operatorname {a2} \,p^{2}+\operatorname {a1} p +\operatorname {a0} \right ) x}{\operatorname {b1} p +\operatorname {b0}}-\frac {\operatorname {c2} \,p^{2}+\operatorname {c1} p +\operatorname {c0}}{\operatorname {b1} p +\operatorname {b0}}\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 {-\operatorname {a2} \,p^{2}-\operatorname {a1} p -\operatorname {a0}}{\operatorname {b1} p +\operatorname {b0}}\\ g &= \frac {-\operatorname {c2} \,p^{2}-\operatorname {c1} p -\operatorname {c0}}{\operatorname {b1} p +\operatorname {b0}} \end {align*}
Hence (2) becomes \begin {align*} p -\frac {-\operatorname {a2} \,p^{2}-\operatorname {a1} p -\operatorname {a0}}{\operatorname {b1} p +\operatorname {b0}} = \left (x \left (\frac {-2 \operatorname {a2} p -\operatorname {a1}}{\operatorname {b1} p +\operatorname {b0}}-\frac {\left (-\operatorname {a2} \,p^{2}-\operatorname {a1} p -\operatorname {a0} \right ) \operatorname {b1}}{\left (\operatorname {b1} p +\operatorname {b0} \right )^{2}}\right )+\frac {-2 \operatorname {c2} p -\operatorname {c1}}{\operatorname {b1} p +\operatorname {b0}}-\frac {\left (-\operatorname {c2} \,p^{2}-\operatorname {c1} p -\operatorname {c0} \right ) \operatorname {b1}}{\left (\operatorname {b1} p +\operatorname {b0} \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 {-\operatorname {a2} \,p^{2}-\operatorname {a1} p -\operatorname {a0}}{\operatorname {b1} p +\operatorname {b0}} = 0 \end {align*}
Solving for \(p\) from the above gives \begin {align*} p&=\frac {-\operatorname {a1} -\operatorname {b0} +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}}{2 \operatorname {a2} +2 \operatorname {b1}}\\ p&=-\frac {\operatorname {a1} +\operatorname {b0} +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}}{2 \left (\operatorname {a2} +\operatorname {b1} \right )} \end {align*}
Substituting these in (1A) gives \begin {align*} y&=\frac {-\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a1} \operatorname {b1} x +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a2} \operatorname {b0} x -2 \operatorname {a0} \operatorname {a2} \operatorname {b1} x -2 \operatorname {a0} \,\operatorname {b1}^{2} x +\operatorname {a1}^{2} \operatorname {b1} x -\operatorname {a1} \operatorname {a2} \operatorname {b0} x +\operatorname {a1} \operatorname {b0} \operatorname {b1} x -\operatorname {a2} \,\operatorname {b0}^{2} x +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a1} \operatorname {c2} -\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a2} \operatorname {c1} +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b0} \operatorname {c2} -\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b1} \operatorname {c1} +2 \operatorname {a0} \operatorname {a2} \operatorname {c2} +2 \operatorname {a0} \operatorname {b1} \operatorname {c2} -\operatorname {a1}^{2} \operatorname {c2} +\operatorname {a1} \operatorname {a2} \operatorname {c1} -2 \operatorname {a1} \operatorname {b0} \operatorname {c2} +\operatorname {a1} \operatorname {b1} \operatorname {c1} -2 \operatorname {a2}^{2} \operatorname {c0} +\operatorname {a2} \operatorname {b0} \operatorname {c1} -4 \operatorname {a2} \operatorname {b1} \operatorname {c0} -\operatorname {b0}^{2} \operatorname {c2} +\operatorname {b0} \operatorname {b1} \operatorname {c1} -2 \operatorname {b1}^{2} \operatorname {c0}}{\operatorname {a2} \sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b1} -\operatorname {a2} \operatorname {a1} \operatorname {b1} +2 \operatorname {a2}^{2} \operatorname {b0} +3 \operatorname {a2} \operatorname {b0} \operatorname {b1} +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b1}^{2}-\operatorname {a1} \,\operatorname {b1}^{2}+\operatorname {b0} \,\operatorname {b1}^{2}}\\ y&=\frac {-\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a1} \operatorname {b1} x +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a2} \operatorname {b0} x +2 \operatorname {a0} \operatorname {a2} \operatorname {b1} x +2 \operatorname {a0} \,\operatorname {b1}^{2} x -\operatorname {a1}^{2} \operatorname {b1} x +\operatorname {a1} \operatorname {a2} \operatorname {b0} x -\operatorname {a1} \operatorname {b0} \operatorname {b1} x +\operatorname {a2} \,\operatorname {b0}^{2} x +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a1} \operatorname {c2} -\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a2} \operatorname {c1} +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b0} \operatorname {c2} -\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b1} \operatorname {c1} -2 \operatorname {a0} \operatorname {a2} \operatorname {c2} -2 \operatorname {a0} \operatorname {b1} \operatorname {c2} +\operatorname {a1}^{2} \operatorname {c2} -\operatorname {a1} \operatorname {a2} \operatorname {c1} +2 \operatorname {a1} \operatorname {b0} \operatorname {c2} -\operatorname {a1} \operatorname {b1} \operatorname {c1} +2 \operatorname {a2}^{2} \operatorname {c0} -\operatorname {a2} \operatorname {b0} \operatorname {c1} +4 \operatorname {a2} \operatorname {b1} \operatorname {c0} +\operatorname {b0}^{2} \operatorname {c2} -\operatorname {b0} \operatorname {b1} \operatorname {c1} +2 \operatorname {b1}^{2} \operatorname {c0}}{\operatorname {a2} \sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b1} +\operatorname {a2} \operatorname {a1} \operatorname {b1} -2 \operatorname {a2}^{2} \operatorname {b0} -3 \operatorname {a2} \operatorname {b0} \operatorname {b1} +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b1}^{2}+\operatorname {a1} \,\operatorname {b1}^{2}-\operatorname {b0} \,\operatorname {b1}^{2}} \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 {-\operatorname {a2} p \left (x \right )^{2}-\operatorname {a1} p \left (x \right )-\operatorname {a0}}{\operatorname {b1} p \left (x \right )+\operatorname {b0}}}{x \left (\frac {-2 \operatorname {a2} p \left (x \right )-\operatorname {a1}}{\operatorname {b1} p \left (x \right )+\operatorname {b0}}-\frac {\left (-\operatorname {a2} p \left (x \right )^{2}-\operatorname {a1} p \left (x \right )-\operatorname {a0} \right ) \operatorname {b1}}{\left (\operatorname {b1} p \left (x \right )+\operatorname {b0} \right )^{2}}\right )+\frac {-2 \operatorname {c2} p \left (x \right )-\operatorname {c1}}{\operatorname {b1} p \left (x \right )+\operatorname {b0}}-\frac {\left (-\operatorname {c2} p \left (x \right )^{2}-\operatorname {c1} p \left (x \right )-\operatorname {c0} \right ) \operatorname {b1}}{\left (\operatorname {b1} p \left (x \right )+\operatorname {b0} \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 \operatorname {a2} p -\operatorname {a1}}{\operatorname {b1} p +\operatorname {b0}}-\frac {\left (-\operatorname {a2} \,p^{2}-\operatorname {a1} p -\operatorname {a0} \right ) \operatorname {b1}}{\left (\operatorname {b1} p +\operatorname {b0} \right )^{2}}\right )+\frac {-2 \operatorname {c2} p -\operatorname {c1}}{\operatorname {b1} p +\operatorname {b0}}-\frac {\left (-\operatorname {c2} \,p^{2}-\operatorname {c1} p -\operatorname {c0} \right ) \operatorname {b1}}{\left (\operatorname {b1} p +\operatorname {b0} \right )^{2}}}{p -\frac {-\operatorname {a2} \,p^{2}-\operatorname {a1} p -\operatorname {a0}}{\operatorname {b1} p +\operatorname {b0}}}\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 {-\operatorname {a2} \,p^{2} \operatorname {b1} -2 \operatorname {a2} p \operatorname {b0} +\operatorname {a0} \operatorname {b1} -\operatorname {a1} \operatorname {b0}}{\left (\operatorname {a2} \,p^{2}+\operatorname {b1} \,p^{2}+\operatorname {a1} p +\operatorname {b0} p +\operatorname {a0} \right ) \left (\operatorname {b1} p +\operatorname {b0} \right )}\\ q(p) &=\frac {-\operatorname {b1} \operatorname {c2} \,p^{2}-2 \operatorname {b0} \operatorname {c2} p -\operatorname {b0} \operatorname {c1} +\operatorname {c0} \operatorname {b1}}{\left (\operatorname {a2} \,p^{2}+\operatorname {b1} \,p^{2}+\operatorname {a1} p +\operatorname {b0} p +\operatorname {a0} \right ) \left (\operatorname {b1} p +\operatorname {b0} \right )} \end {align*}
Hence the ode is \begin {align*} \frac {d}{d p}x \left (p \right )-\frac {\left (-\operatorname {a2} \,p^{2} \operatorname {b1} -2 \operatorname {a2} p \operatorname {b0} +\operatorname {a0} \operatorname {b1} -\operatorname {a1} \operatorname {b0} \right ) x \left (p \right )}{\left (\operatorname {a2} \,p^{2}+\operatorname {b1} \,p^{2}+\operatorname {a1} p +\operatorname {b0} p +\operatorname {a0} \right ) \left (\operatorname {b1} p +\operatorname {b0} \right )} = \frac {-\operatorname {b1} \operatorname {c2} \,p^{2}-2 \operatorname {b0} \operatorname {c2} p -\operatorname {b0} \operatorname {c1} +\operatorname {c0} \operatorname {b1}}{\left (\operatorname {a2} \,p^{2}+\operatorname {b1} \,p^{2}+\operatorname {a1} p +\operatorname {b0} p +\operatorname {a0} \right ) \left (\operatorname {b1} p +\operatorname {b0} \right )} \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int -\frac {-\operatorname {a2} \,p^{2} \operatorname {b1} -2 \operatorname {a2} p \operatorname {b0} +\operatorname {a0} \operatorname {b1} -\operatorname {a1} \operatorname {b0}}{\left (\operatorname {a2} \,p^{2}+\operatorname {b1} \,p^{2}+\operatorname {a1} p +\operatorname {b0} p +\operatorname {a0} \right ) \left (\operatorname {b1} p +\operatorname {b0} \right )}d p} \\ &= {\mathrm e}^{-\frac {\left (-2 \operatorname {a2} -\operatorname {b1} \right ) \ln \left (\operatorname {a2} \,p^{2}+\operatorname {b1} \,p^{2}+\operatorname {a1} p +\operatorname {b0} p +\operatorname {a0} \right )}{2 \left (\operatorname {a2} +\operatorname {b1} \right )}-\frac {2 \left (-\operatorname {a1} -\frac {\left (-2 \operatorname {a2} -\operatorname {b1} \right ) \left (\operatorname {a1} +\operatorname {b0} \right )}{2 \left (\operatorname {a2} +\operatorname {b1} \right )}\right ) \arctan \left (\frac {2 \left (\operatorname {a2} +\operatorname {b1} \right ) p +\operatorname {a1} +\operatorname {b0}}{\sqrt {4 \operatorname {a2} \operatorname {a0} +4 \operatorname {a0} \operatorname {b1} -\operatorname {a1}^{2}-2 \operatorname {a1} \operatorname {b0} -\operatorname {b0}^{2}}}\right )}{\sqrt {4 \operatorname {a2} \operatorname {a0} +4 \operatorname {a0} \operatorname {b1} -\operatorname {a1}^{2}-2 \operatorname {a1} \operatorname {b0} -\operatorname {b0}^{2}}}-\ln \left (\operatorname {b1} p +\operatorname {b0} \right )} \\ \end{align*} Which simplifies to \[ \mu = \frac {\left (\left (\operatorname {a2} +\operatorname {b1} \right ) p^{2}+\left (\operatorname {a1} +\operatorname {b0} \right ) p +\operatorname {a0} \right )^{\frac {2 \operatorname {a2} +\operatorname {b1}}{2 \operatorname {a2} +2 \operatorname {b1}}} {\mathrm e}^{\frac {\arctan \left (\frac {2 \operatorname {a2} p +2 \operatorname {b1} p +\operatorname {a1} +\operatorname {b0}}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}}\right ) \left (\left (\operatorname {a1} -\operatorname {b0} \right ) \operatorname {b1} -2 \operatorname {a2} \operatorname {b0} \right )}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}\, \left (\operatorname {a2} +\operatorname {b1} \right )}}}{\operatorname {b1} p +\operatorname {b0}} \] The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}p}}\left ( \mu x\right ) &= \left (\mu \right ) \left (\frac {-\operatorname {b1} \operatorname {c2} \,p^{2}-2 \operatorname {b0} \operatorname {c2} p -\operatorname {b0} \operatorname {c1} +\operatorname {c0} \operatorname {b1}}{\left (\operatorname {a2} \,p^{2}+\operatorname {b1} \,p^{2}+\operatorname {a1} p +\operatorname {b0} p +\operatorname {a0} \right ) \left (\operatorname {b1} p +\operatorname {b0} \right )}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}p}} \left (\frac {\left (\left (\operatorname {a2} +\operatorname {b1} \right ) p^{2}+\left (\operatorname {a1} +\operatorname {b0} \right ) p +\operatorname {a0} \right )^{\frac {2 \operatorname {a2} +\operatorname {b1}}{2 \operatorname {a2} +2 \operatorname {b1}}} {\mathrm e}^{\frac {\arctan \left (\frac {2 \operatorname {a2} p +2 \operatorname {b1} p +\operatorname {a1} +\operatorname {b0}}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}}\right ) \left (\left (\operatorname {a1} -\operatorname {b0} \right ) \operatorname {b1} -2 \operatorname {a2} \operatorname {b0} \right )}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}\, \left (\operatorname {a2} +\operatorname {b1} \right )}} x}{\operatorname {b1} p +\operatorname {b0}}\right ) &= \left (\frac {\left (\left (\operatorname {a2} +\operatorname {b1} \right ) p^{2}+\left (\operatorname {a1} +\operatorname {b0} \right ) p +\operatorname {a0} \right )^{\frac {2 \operatorname {a2} +\operatorname {b1}}{2 \operatorname {a2} +2 \operatorname {b1}}} {\mathrm e}^{\frac {\arctan \left (\frac {2 \operatorname {a2} p +2 \operatorname {b1} p +\operatorname {a1} +\operatorname {b0}}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}}\right ) \left (\left (\operatorname {a1} -\operatorname {b0} \right ) \operatorname {b1} -2 \operatorname {a2} \operatorname {b0} \right )}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}\, \left (\operatorname {a2} +\operatorname {b1} \right )}}}{\operatorname {b1} p +\operatorname {b0}}\right ) \left (\frac {-\operatorname {b1} \operatorname {c2} \,p^{2}-2 \operatorname {b0} \operatorname {c2} p -\operatorname {b0} \operatorname {c1} +\operatorname {c0} \operatorname {b1}}{\left (\operatorname {a2} \,p^{2}+\operatorname {b1} \,p^{2}+\operatorname {a1} p +\operatorname {b0} p +\operatorname {a0} \right ) \left (\operatorname {b1} p +\operatorname {b0} \right )}\right )\\ \mathrm {d} \left (\frac {\left (\left (\operatorname {a2} +\operatorname {b1} \right ) p^{2}+\left (\operatorname {a1} +\operatorname {b0} \right ) p +\operatorname {a0} \right )^{\frac {2 \operatorname {a2} +\operatorname {b1}}{2 \operatorname {a2} +2 \operatorname {b1}}} {\mathrm e}^{\frac {\arctan \left (\frac {2 \operatorname {a2} p +2 \operatorname {b1} p +\operatorname {a1} +\operatorname {b0}}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}}\right ) \left (\left (\operatorname {a1} -\operatorname {b0} \right ) \operatorname {b1} -2 \operatorname {a2} \operatorname {b0} \right )}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}\, \left (\operatorname {a2} +\operatorname {b1} \right )}} x}{\operatorname {b1} p +\operatorname {b0}}\right ) &= \left (-\frac {{\mathrm e}^{\frac {\arctan \left (\frac {2 \operatorname {a2} p +2 \operatorname {b1} p +\operatorname {a1} +\operatorname {b0}}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}}\right ) \left (\left (\operatorname {a1} -\operatorname {b0} \right ) \operatorname {b1} -2 \operatorname {a2} \operatorname {b0} \right )}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}\, \left (\operatorname {a2} +\operatorname {b1} \right )}} \left (\left (2 \operatorname {c2} p +\operatorname {c1} \right ) \operatorname {b0} +\operatorname {b1} \left (\operatorname {c2} \,p^{2}-\operatorname {c0} \right )\right ) \left (\left (\operatorname {a2} +\operatorname {b1} \right ) p^{2}+\left (\operatorname {a1} +\operatorname {b0} \right ) p +\operatorname {a0} \right )^{-\frac {\operatorname {b1}}{2 \operatorname {a2} +2 \operatorname {b1}}}}{\left (\operatorname {b1} p +\operatorname {b0} \right )^{2}}\right )\, \mathrm {d} p \end {align*}
Integrating gives \begin {align*} \frac {\left (\left (\operatorname {a2} +\operatorname {b1} \right ) p^{2}+\left (\operatorname {a1} +\operatorname {b0} \right ) p +\operatorname {a0} \right )^{\frac {2 \operatorname {a2} +\operatorname {b1}}{2 \operatorname {a2} +2 \operatorname {b1}}} {\mathrm e}^{\frac {\arctan \left (\frac {2 \operatorname {a2} p +2 \operatorname {b1} p +\operatorname {a1} +\operatorname {b0}}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}}\right ) \left (\left (\operatorname {a1} -\operatorname {b0} \right ) \operatorname {b1} -2 \operatorname {a2} \operatorname {b0} \right )}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}\, \left (\operatorname {a2} +\operatorname {b1} \right )}} x}{\operatorname {b1} p +\operatorname {b0}} &= \int {-\frac {{\mathrm e}^{\frac {\arctan \left (\frac {2 \operatorname {a2} p +2 \operatorname {b1} p +\operatorname {a1} +\operatorname {b0}}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}}\right ) \left (\left (\operatorname {a1} -\operatorname {b0} \right ) \operatorname {b1} -2 \operatorname {a2} \operatorname {b0} \right )}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}\, \left (\operatorname {a2} +\operatorname {b1} \right )}} \left (\left (2 \operatorname {c2} p +\operatorname {c1} \right ) \operatorname {b0} +\operatorname {b1} \left (\operatorname {c2} \,p^{2}-\operatorname {c0} \right )\right ) \left (\left (\operatorname {a2} +\operatorname {b1} \right ) p^{2}+\left (\operatorname {a1} +\operatorname {b0} \right ) p +\operatorname {a0} \right )^{-\frac {\operatorname {b1}}{2 \operatorname {a2} +2 \operatorname {b1}}}}{\left (\operatorname {b1} p +\operatorname {b0} \right )^{2}}\,\mathrm {d} p}\\ \frac {\left (\left (\operatorname {a2} +\operatorname {b1} \right ) p^{2}+\left (\operatorname {a1} +\operatorname {b0} \right ) p +\operatorname {a0} \right )^{\frac {2 \operatorname {a2} +\operatorname {b1}}{2 \operatorname {a2} +2 \operatorname {b1}}} {\mathrm e}^{\frac {\arctan \left (\frac {2 \operatorname {a2} p +2 \operatorname {b1} p +\operatorname {a1} +\operatorname {b0}}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}}\right ) \left (\left (\operatorname {a1} -\operatorname {b0} \right ) \operatorname {b1} -2 \operatorname {a2} \operatorname {b0} \right )}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}\, \left (\operatorname {a2} +\operatorname {b1} \right )}} x}{\operatorname {b1} p +\operatorname {b0}} &= \int -\frac {{\mathrm e}^{\frac {\arctan \left (\frac {2 \operatorname {a2} p +2 \operatorname {b1} p +\operatorname {a1} +\operatorname {b0}}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}}\right ) \left (\left (\operatorname {a1} -\operatorname {b0} \right ) \operatorname {b1} -2 \operatorname {a2} \operatorname {b0} \right )}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}\, \left (\operatorname {a2} +\operatorname {b1} \right )}} \left (\left (2 \operatorname {c2} p +\operatorname {c1} \right ) \operatorname {b0} +\operatorname {b1} \left (\operatorname {c2} \,p^{2}-\operatorname {c0} \right )\right ) \left (\left (\operatorname {a2} +\operatorname {b1} \right ) p^{2}+\left (\operatorname {a1} +\operatorname {b0} \right ) p +\operatorname {a0} \right )^{-\frac {\operatorname {b1}}{2 \operatorname {a2} +2 \operatorname {b1}}}}{\left (\operatorname {b1} p +\operatorname {b0} \right )^{2}}d p + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu =\frac {\left (\left (\operatorname {a2} +\operatorname {b1} \right ) p^{2}+\left (\operatorname {a1} +\operatorname {b0} \right ) p +\operatorname {a0} \right )^{\frac {2 \operatorname {a2} +\operatorname {b1}}{2 \operatorname {a2} +2 \operatorname {b1}}} {\mathrm e}^{\frac {\arctan \left (\frac {2 \operatorname {a2} p +2 \operatorname {b1} p +\operatorname {a1} +\operatorname {b0}}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}}\right ) \left (\left (\operatorname {a1} -\operatorname {b0} \right ) \operatorname {b1} -2 \operatorname {a2} \operatorname {b0} \right )}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}\, \left (\operatorname {a2} +\operatorname {b1} \right )}}}{\operatorname {b1} p +\operatorname {b0}}\) results in \begin {align*} x \left (p \right ) &= \left (\operatorname {b1} p +\operatorname {b0} \right ) \left (\left (\operatorname {a2} +\operatorname {b1} \right ) p^{2}+\left (\operatorname {a1} +\operatorname {b0} \right ) p +\operatorname {a0} \right )^{\frac {-2 \operatorname {a2} -\operatorname {b1}}{2 \operatorname {a2} +2 \operatorname {b1}}} {\mathrm e}^{-\frac {\arctan \left (\frac {2 \operatorname {a2} p +2 \operatorname {b1} p +\operatorname {a1} +\operatorname {b0}}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}}\right ) \left (\left (\operatorname {a1} -\operatorname {b0} \right ) \operatorname {b1} -2 \operatorname {a2} \operatorname {b0} \right )}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}\, \left (\operatorname {a2} +\operatorname {b1} \right )}} \left (\int -\frac {{\mathrm e}^{\frac {\arctan \left (\frac {2 \operatorname {a2} p +2 \operatorname {b1} p +\operatorname {a1} +\operatorname {b0}}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}}\right ) \left (\left (\operatorname {a1} -\operatorname {b0} \right ) \operatorname {b1} -2 \operatorname {a2} \operatorname {b0} \right )}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}\, \left (\operatorname {a2} +\operatorname {b1} \right )}} \left (\left (2 \operatorname {c2} p +\operatorname {c1} \right ) \operatorname {b0} +\operatorname {b1} \left (\operatorname {c2} \,p^{2}-\operatorname {c0} \right )\right ) \left (\left (\operatorname {a2} +\operatorname {b1} \right ) p^{2}+\left (\operatorname {a1} +\operatorname {b0} \right ) p +\operatorname {a0} \right )^{-\frac {\operatorname {b1}}{2 \operatorname {a2} +2 \operatorname {b1}}}}{\left (\operatorname {b1} p +\operatorname {b0} \right )^{2}}d p \right )+c_{1} \left (\operatorname {b1} p +\operatorname {b0} \right ) \left (\left (\operatorname {a2} +\operatorname {b1} \right ) p^{2}+\left (\operatorname {a1} +\operatorname {b0} \right ) p +\operatorname {a0} \right )^{\frac {-2 \operatorname {a2} -\operatorname {b1}}{2 \operatorname {a2} +2 \operatorname {b1}}} {\mathrm e}^{-\frac {\arctan \left (\frac {2 \operatorname {a2} p +2 \operatorname {b1} p +\operatorname {a1} +\operatorname {b0}}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}}\right ) \left (\left (\operatorname {a1} -\operatorname {b0} \right ) \operatorname {b1} -2 \operatorname {a2} \operatorname {b0} \right )}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}\, \left (\operatorname {a2} +\operatorname {b1} \right )}} \end {align*}
which simplifies to \begin {align*} x \left (p \right ) &= \left (\left (\operatorname {a2} +\operatorname {b1} \right ) p^{2}+\left (\operatorname {a1} +\operatorname {b0} \right ) p +\operatorname {a0} \right )^{\frac {-2 \operatorname {a2} -\operatorname {b1}}{2 \operatorname {a2} +2 \operatorname {b1}}} {\mathrm e}^{-\frac {\arctan \left (\frac {2 \operatorname {a2} p +2 \operatorname {b1} p +\operatorname {a1} +\operatorname {b0}}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}}\right ) \left (\left (\operatorname {a1} -\operatorname {b0} \right ) \operatorname {b1} -2 \operatorname {a2} \operatorname {b0} \right )}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}\, \left (\operatorname {a2} +\operatorname {b1} \right )}} \left (c_{1} -\left (\int \frac {{\mathrm e}^{\frac {\arctan \left (\frac {2 \operatorname {a2} p +2 \operatorname {b1} p +\operatorname {a1} +\operatorname {b0}}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}}\right ) \left (\left (\operatorname {a1} -\operatorname {b0} \right ) \operatorname {b1} -2 \operatorname {a2} \operatorname {b0} \right )}{\sqrt {\left (4 \operatorname {a2} +4 \operatorname {b1} \right ) \operatorname {a0} -\left (\operatorname {a1} +\operatorname {b0} \right )^{2}}\, \left (\operatorname {a2} +\operatorname {b1} \right )}} \left (\left (2 \operatorname {c2} p +\operatorname {c1} \right ) \operatorname {b0} +\operatorname {b1} \left (\operatorname {c2} \,p^{2}-\operatorname {c0} \right )\right ) \left (\left (\operatorname {a2} +\operatorname {b1} \right ) p^{2}+\left (\operatorname {a1} +\operatorname {b0} \right ) p +\operatorname {a0} \right )^{-\frac {\operatorname {b1}}{2 \operatorname {a2} +2 \operatorname {b1}}}}{\left (\operatorname {b1} p +\operatorname {b0} \right )^{2}}d p \right )\right ) \left (\operatorname {b1} p +\operatorname {b0} \right ) \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 {-\operatorname {a1} x -\operatorname {b1} y-\operatorname {c1} +\sqrt {y^{2} \operatorname {b1}^{2}+2 y \operatorname {a1} \operatorname {b1} x -4 y \operatorname {a2} \operatorname {b0} x -4 \operatorname {a0} \operatorname {a2} \,x^{2}+\operatorname {a1}^{2} x^{2}-4 y \operatorname {b0} \operatorname {c2} +2 y \operatorname {b1} \operatorname {c1} -4 \operatorname {a0} \operatorname {c2} x +2 \operatorname {a1} \operatorname {c1} x -4 \operatorname {a2} \operatorname {c0} x -4 \operatorname {c0} \operatorname {c2} +\operatorname {c1}^{2}}}{2 \operatorname {a2} x +2 \operatorname {c2}}\\ p&=-\frac {\operatorname {b1} y+\operatorname {a1} x +\sqrt {y^{2} \operatorname {b1}^{2}+2 y \operatorname {a1} \operatorname {b1} x -4 y \operatorname {a2} \operatorname {b0} x -4 \operatorname {a0} \operatorname {a2} \,x^{2}+\operatorname {a1}^{2} x^{2}-4 y \operatorname {b0} \operatorname {c2} +2 y \operatorname {b1} \operatorname {c1} -4 \operatorname {a0} \operatorname {c2} x +2 \operatorname {a1} \operatorname {c1} x -4 \operatorname {a2} \operatorname {c0} x -4 \operatorname {c0} \operatorname {c2} +\operatorname {c1}^{2}}+\operatorname {c1}}{2 \left (\operatorname {a2} x +\operatorname {c2} \right )} \end {align*}
Substituting the above in the solution for \(x\) found above gives \begin{align*} x&=\text {Expression too large to display} \\ x&=\text {Expression too large to display} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {-\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a1} \operatorname {b1} x +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a2} \operatorname {b0} x -2 \operatorname {a0} \operatorname {a2} \operatorname {b1} x -2 \operatorname {a0} \,\operatorname {b1}^{2} x +\operatorname {a1}^{2} \operatorname {b1} x -\operatorname {a1} \operatorname {a2} \operatorname {b0} x +\operatorname {a1} \operatorname {b0} \operatorname {b1} x -\operatorname {a2} \,\operatorname {b0}^{2} x +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a1} \operatorname {c2} -\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a2} \operatorname {c1} +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b0} \operatorname {c2} -\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b1} \operatorname {c1} +2 \operatorname {a0} \operatorname {a2} \operatorname {c2} +2 \operatorname {a0} \operatorname {b1} \operatorname {c2} -\operatorname {a1}^{2} \operatorname {c2} +\operatorname {a1} \operatorname {a2} \operatorname {c1} -2 \operatorname {a1} \operatorname {b0} \operatorname {c2} +\operatorname {a1} \operatorname {b1} \operatorname {c1} -2 \operatorname {a2}^{2} \operatorname {c0} +\operatorname {a2} \operatorname {b0} \operatorname {c1} -4 \operatorname {a2} \operatorname {b1} \operatorname {c0} -\operatorname {b0}^{2} \operatorname {c2} +\operatorname {b0} \operatorname {b1} \operatorname {c1} -2 \operatorname {b1}^{2} \operatorname {c0}}{\operatorname {a2} \sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b1} -\operatorname {a2} \operatorname {a1} \operatorname {b1} +2 \operatorname {a2}^{2} \operatorname {b0} +3 \operatorname {a2} \operatorname {b0} \operatorname {b1} +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b1}^{2}-\operatorname {a1} \,\operatorname {b1}^{2}+\operatorname {b0} \,\operatorname {b1}^{2}} \\ \tag{2} y &= \frac {-\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a1} \operatorname {b1} x +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a2} \operatorname {b0} x +2 \operatorname {a0} \operatorname {a2} \operatorname {b1} x +2 \operatorname {a0} \,\operatorname {b1}^{2} x -\operatorname {a1}^{2} \operatorname {b1} x +\operatorname {a1} \operatorname {a2} \operatorname {b0} x -\operatorname {a1} \operatorname {b0} \operatorname {b1} x +\operatorname {a2} \,\operatorname {b0}^{2} x +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a1} \operatorname {c2} -\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a2} \operatorname {c1} +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b0} \operatorname {c2} -\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b1} \operatorname {c1} -2 \operatorname {a0} \operatorname {a2} \operatorname {c2} -2 \operatorname {a0} \operatorname {b1} \operatorname {c2} +\operatorname {a1}^{2} \operatorname {c2} -\operatorname {a1} \operatorname {a2} \operatorname {c1} +2 \operatorname {a1} \operatorname {b0} \operatorname {c2} -\operatorname {a1} \operatorname {b1} \operatorname {c1} +2 \operatorname {a2}^{2} \operatorname {c0} -\operatorname {a2} \operatorname {b0} \operatorname {c1} +4 \operatorname {a2} \operatorname {b1} \operatorname {c0} +\operatorname {b0}^{2} \operatorname {c2} -\operatorname {b0} \operatorname {b1} \operatorname {c1} +2 \operatorname {b1}^{2} \operatorname {c0}}{\operatorname {a2} \sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b1} +\operatorname {a2} \operatorname {a1} \operatorname {b1} -2 \operatorname {a2}^{2} \operatorname {b0} -3 \operatorname {a2} \operatorname {b0} \operatorname {b1} +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b1}^{2}+\operatorname {a1} \,\operatorname {b1}^{2}-\operatorname {b0} \,\operatorname {b1}^{2}} \\ \tag{3} \text {Expression too large to display} \\ \tag{4} \text {Expression too large to display} \\ \end{align*}
Verification of solutions
\[ y = \frac {-\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a1} \operatorname {b1} x +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a2} \operatorname {b0} x -2 \operatorname {a0} \operatorname {a2} \operatorname {b1} x -2 \operatorname {a0} \,\operatorname {b1}^{2} x +\operatorname {a1}^{2} \operatorname {b1} x -\operatorname {a1} \operatorname {a2} \operatorname {b0} x +\operatorname {a1} \operatorname {b0} \operatorname {b1} x -\operatorname {a2} \,\operatorname {b0}^{2} x +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a1} \operatorname {c2} -\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a2} \operatorname {c1} +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b0} \operatorname {c2} -\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b1} \operatorname {c1} +2 \operatorname {a0} \operatorname {a2} \operatorname {c2} +2 \operatorname {a0} \operatorname {b1} \operatorname {c2} -\operatorname {a1}^{2} \operatorname {c2} +\operatorname {a1} \operatorname {a2} \operatorname {c1} -2 \operatorname {a1} \operatorname {b0} \operatorname {c2} +\operatorname {a1} \operatorname {b1} \operatorname {c1} -2 \operatorname {a2}^{2} \operatorname {c0} +\operatorname {a2} \operatorname {b0} \operatorname {c1} -4 \operatorname {a2} \operatorname {b1} \operatorname {c0} -\operatorname {b0}^{2} \operatorname {c2} +\operatorname {b0} \operatorname {b1} \operatorname {c1} -2 \operatorname {b1}^{2} \operatorname {c0}}{\operatorname {a2} \sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b1} -\operatorname {a2} \operatorname {a1} \operatorname {b1} +2 \operatorname {a2}^{2} \operatorname {b0} +3 \operatorname {a2} \operatorname {b0} \operatorname {b1} +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b1}^{2}-\operatorname {a1} \,\operatorname {b1}^{2}+\operatorname {b0} \,\operatorname {b1}^{2}} \] Verified OK.
\[ y = \frac {-\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a1} \operatorname {b1} x +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a2} \operatorname {b0} x +2 \operatorname {a0} \operatorname {a2} \operatorname {b1} x +2 \operatorname {a0} \,\operatorname {b1}^{2} x -\operatorname {a1}^{2} \operatorname {b1} x +\operatorname {a1} \operatorname {a2} \operatorname {b0} x -\operatorname {a1} \operatorname {b0} \operatorname {b1} x +\operatorname {a2} \,\operatorname {b0}^{2} x +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a1} \operatorname {c2} -\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {a2} \operatorname {c1} +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b0} \operatorname {c2} -\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b1} \operatorname {c1} -2 \operatorname {a0} \operatorname {a2} \operatorname {c2} -2 \operatorname {a0} \operatorname {b1} \operatorname {c2} +\operatorname {a1}^{2} \operatorname {c2} -\operatorname {a1} \operatorname {a2} \operatorname {c1} +2 \operatorname {a1} \operatorname {b0} \operatorname {c2} -\operatorname {a1} \operatorname {b1} \operatorname {c1} +2 \operatorname {a2}^{2} \operatorname {c0} -\operatorname {a2} \operatorname {b0} \operatorname {c1} +4 \operatorname {a2} \operatorname {b1} \operatorname {c0} +\operatorname {b0}^{2} \operatorname {c2} -\operatorname {b0} \operatorname {b1} \operatorname {c1} +2 \operatorname {b1}^{2} \operatorname {c0}}{\operatorname {a2} \sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b1} +\operatorname {a2} \operatorname {a1} \operatorname {b1} -2 \operatorname {a2}^{2} \operatorname {b0} -3 \operatorname {a2} \operatorname {b0} \operatorname {b1} +\sqrt {-4 \operatorname {a2} \operatorname {a0} -4 \operatorname {a0} \operatorname {b1} +\operatorname {a1}^{2}+2 \operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}\, \operatorname {b1}^{2}+\operatorname {a1} \,\operatorname {b1}^{2}-\operatorname {b0} \,\operatorname {b1}^{2}} \] Verified OK.
\[ \text {Expression too large to display} \] 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 (\mathit {a2} x +\mathit {c2} \right ) {y^{\prime }}^{2}+\left (\mathit {a1} x +\mathit {b1} y+\mathit {c1} \right ) y^{\prime }+\mathit {b0} y=-\mathit {a0} x -\mathit {c0} \\ \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 {\mathit {b1} y+\mathit {a1} x -\sqrt {y^{2} \mathit {b1}^{2}+2 y \mathit {a1} \mathit {b1} x -4 y \mathit {a2} \mathit {b0} x -4 \mathit {a0} \mathit {a2} \,x^{2}+\mathit {a1}^{2} x^{2}-4 y \mathit {b0} \mathit {c2} +2 y \mathit {b1} \mathit {c1} -4 \mathit {a0} \mathit {c2} x +2 \mathit {a1} \mathit {c1} x -4 \mathit {a2} \mathit {c0} x -4 \mathit {c0} \mathit {c2} +\mathit {c1}^{2}}+\mathit {c1}}{2 \left (\mathit {a2} x +\mathit {c2} \right )}, y^{\prime }=-\frac {\mathit {b1} y+\mathit {a1} x +\sqrt {y^{2} \mathit {b1}^{2}+2 y \mathit {a1} \mathit {b1} x -4 y \mathit {a2} \mathit {b0} x -4 \mathit {a0} \mathit {a2} \,x^{2}+\mathit {a1}^{2} x^{2}-4 y \mathit {b0} \mathit {c2} +2 y \mathit {b1} \mathit {c1} -4 \mathit {a0} \mathit {c2} x +2 \mathit {a1} \mathit {c1} x -4 \mathit {a2} \mathit {c0} x -4 \mathit {c0} \mathit {c2} +\mathit {c1}^{2}}+\mathit {c1}}{2 \left (\mathit {a2} x +\mathit {c2} \right )}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\mathit {b1} y+\mathit {a1} x -\sqrt {y^{2} \mathit {b1}^{2}+2 y \mathit {a1} \mathit {b1} x -4 y \mathit {a2} \mathit {b0} x -4 \mathit {a0} \mathit {a2} \,x^{2}+\mathit {a1}^{2} x^{2}-4 y \mathit {b0} \mathit {c2} +2 y \mathit {b1} \mathit {c1} -4 \mathit {a0} \mathit {c2} x +2 \mathit {a1} \mathit {c1} x -4 \mathit {a2} \mathit {c0} x -4 \mathit {c0} \mathit {c2} +\mathit {c1}^{2}}+\mathit {c1}}{2 \left (\mathit {a2} x +\mathit {c2} \right )} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\mathit {b1} y+\mathit {a1} x +\sqrt {y^{2} \mathit {b1}^{2}+2 y \mathit {a1} \mathit {b1} x -4 y \mathit {a2} \mathit {b0} x -4 \mathit {a0} \mathit {a2} \,x^{2}+\mathit {a1}^{2} x^{2}-4 y \mathit {b0} \mathit {c2} +2 y \mathit {b1} \mathit {c1} -4 \mathit {a0} \mathit {c2} x +2 \mathit {a1} \mathit {c1} x -4 \mathit {a2} \mathit {c0} x -4 \mathit {c0} \mathit {c2} +\mathit {c1}^{2}}+\mathit {c1}}{2 \left (\mathit {a2} x +\mathit {c2} \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: 0.36 (sec). Leaf size: 1983
dsolve((a2*x+c2)*diff(y(x),x)^2+(a1*x+b1*y(x)+c1)*diff(y(x),x)+a0*x+b0*y(x)+c0 = 0,y(x), singsol=all)
\begin{align*} \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align*}
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[c0 + a0*x + b0*y[x] + (c1 + a1*x + b1*y[x])*y'[x] + (c2 + a2*x)*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
Timed out