29.15.18 problem 426

Internal problem ID [5024]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 15
Problem number : 426
Date solved : Tuesday, March 04, 2025 at 07:43:51 PM
CAS classification : [_quadrature]

\begin{align*} y y^{\prime }&=\operatorname {a0} +\operatorname {a1} y+\operatorname {a2} y^{2} \end{align*}

✓ Maple. Time used: 0.572 (sec). Leaf size: 222

ode:=y(x)*diff(y(x),x) = a0+a1*y(x)+a2*y(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y \left (x \right ) = \frac {4 \tan \left (\operatorname {RootOf}\left (2 c_{1} \operatorname {a2} \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}+2 x \operatorname {a2} \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}+2 \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}\, \ln \left (2\right )-\sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}\, \ln \left (\frac {\sec \left (\textit {\_Z} \right )^{2} \left (4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}\right )}{\operatorname {a2}}\right )+2 \textit {\_Z} \operatorname {a1} \right )\right ) \operatorname {a0} \operatorname {a2} -\tan \left (\operatorname {RootOf}\left (2 c_{1} \operatorname {a2} \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}+2 x \operatorname {a2} \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}+2 \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}\, \ln \left (2\right )-\sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}\, \ln \left (\frac {\sec \left (\textit {\_Z} \right )^{2} \left (4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}\right )}{\operatorname {a2}}\right )+2 \textit {\_Z} \operatorname {a1} \right )\right ) \operatorname {a1}^{2}-\sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}\, \operatorname {a1}}{2 \operatorname {a2} \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}} \]

✓ Mathematica. Time used: 0.437 (sec). Leaf size: 123

ode=y[x] D[y[x],x]==a0+a1 y[x]+a2 y[x]^2; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 4.354 (sec). Leaf size: 192

from sympy import * 
x = symbols("x") 
a0 = symbols("a0") 
a1 = symbols("a1") 
a2 = symbols("a2") 
y = Function("y") 
ode = Eq(-a0 - a1*y(x) - a2*y(x)**2 + y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ - x + \frac {\left (- \frac {a_{1} \sqrt {- 4 a_{0} a_{2} + a_{1}^{2}}}{4 a_{0} a_{2} - a_{1}^{2}} + 1\right ) \log {\left (y{\left (x \right )} + \frac {- 2 a_{0} \left (- \frac {a_{1} \sqrt {- 4 a_{0} a_{2} + a_{1}^{2}}}{4 a_{0} a_{2} - a_{1}^{2}} + 1\right ) + 2 a_{0} + \frac {a_{1}^{2} \left (- \frac {a_{1} \sqrt {- 4 a_{0} a_{2} + a_{1}^{2}}}{4 a_{0} a_{2} - a_{1}^{2}} + 1\right )}{2 a_{2}}}{a_{1}} \right )}}{2 a_{2}} + \frac {\left (\frac {a_{1} \sqrt {- 4 a_{0} a_{2} + a_{1}^{2}}}{4 a_{0} a_{2} - a_{1}^{2}} + 1\right ) \log {\left (y{\left (x \right )} + \frac {- 2 a_{0} \left (\frac {a_{1} \sqrt {- 4 a_{0} a_{2} + a_{1}^{2}}}{4 a_{0} a_{2} - a_{1}^{2}} + 1\right ) + 2 a_{0} + \frac {a_{1}^{2} \left (\frac {a_{1} \sqrt {- 4 a_{0} a_{2} + a_{1}^{2}}}{4 a_{0} a_{2} - a_{1}^{2}} + 1\right )}{2 a_{2}}}{a_{1}} \right )}}{2 a_{2}} = C_{1} \]