Internal problem ID [3171]
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]
\[ \boxed {y y^{\prime }-\operatorname {a0} -\operatorname {a1} y-\operatorname {a2} y^{2}=0} \]
✓ Solution by Maple
Time used: 0.141 (sec). Leaf size: 218
dsolve(y(x)*diff(y(x),x) = a0+a1*y(x)+a2*y(x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {4 \operatorname {a0} \operatorname {a2} \tan \left (\operatorname {RootOf}\left (2 c_{1} \operatorname {a2} \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}+2 x \operatorname {a2} \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}-\sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}\, \ln \left (\frac {4 \operatorname {a0} \operatorname {a2} \tan \left (\textit {\_Z} \right )^{2}-\operatorname {a1}^{2} \tan \left (\textit {\_Z} \right )^{2}+4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}{4 \operatorname {a2}}\right )+2 \textit {\_Z} \operatorname {a1} \right )\right )-\operatorname {a1}^{2} \tan \left (\operatorname {RootOf}\left (2 c_{1} \operatorname {a2} \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}+2 x \operatorname {a2} \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}-\sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}\, \ln \left (\frac {4 \operatorname {a0} \operatorname {a2} \tan \left (\textit {\_Z} \right )^{2}-\operatorname {a1}^{2} \tan \left (\textit {\_Z} \right )^{2}+4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}{4 \operatorname {a2}}\right )+2 \textit {\_Z} \operatorname {a1} \right )\right )-\sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}\, \operatorname {a1}}{2 \operatorname {a2} \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}} \]
✓ Solution by Mathematica
Time used: 0.397 (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 {$\#$1} (\text {$\#$1} \text {a2}+\text {a1})+\text {a0})-\frac {2 \text {a1} \arctan \left (\frac {2 \text {$\#$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*}