Internal problem ID [3172]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 15
Problem number: 426.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {y y^{\prime }-\mathit {a0} -\mathit {a1} y-\mathit {a2} y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.099 (sec). Leaf size: 218
dsolve(y(x)*diff(y(x),x) = a0+a1*y(x)+a2*y(x)^2,y(x), singsol=all)
\[ y \relax (x ) = \frac {4 \mathit {a0} \mathit {a2} \tan \left (\RootOf \left (2 c_{1} \mathit {a2} \sqrt {4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}+2 x \mathit {a2} \sqrt {4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}-\ln \left (\frac {4 \mathit {a0} \mathit {a2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )-\mathit {a1}^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}{4 \mathit {a2}}\right ) \sqrt {4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}+2 \textit {\_Z} \mathit {a1} \right )\right )-\mathit {a1}^{2} \tan \left (\RootOf \left (2 c_{1} \mathit {a2} \sqrt {4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}+2 x \mathit {a2} \sqrt {4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}-\ln \left (\frac {4 \mathit {a0} \mathit {a2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )-\mathit {a1}^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}{4 \mathit {a2}}\right ) \sqrt {4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}+2 \textit {\_Z} \mathit {a1} \right )\right )-\sqrt {4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}\, \mathit {a1}}{2 \sqrt {4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}\, \mathit {a2}} \]
✓ Solution by Mathematica
Time used: 0.396 (sec). Leaf size: 123
DSolve[y[x] y'[x]==a0+a1 y[x]+a2 y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {\log (\text {$\#$1} (\text {$\#$1} \text {a2}+\text {a1})+\text {a0})-\frac {2 \text {a1} \text {ArcTan}\left (\frac {2 \text {$\#$1} \text {a2}+\text {a1}}{\sqrt {4 \text {a0} \text {a2}-\text {a1}^2}}\right )}{\sqrt {4 \text {a0} \text {a2}-\text {a1}^2}}}{2 \text {a2}}\&\right ][x+c_1] \\ y(x)\to \frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}-\text {a1}}{2 \text {a2}} \\ y(x)\to -\frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}+\text {a1}}{2 \text {a2}} \\ \end{align*}