61.3.17 problem 17
Internal
problem
ID
[12101]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
subsection
1.2.3.
Equations
Containing
Exponential
Functions
Problem
number
:
17
Date
solved
:
Tuesday, January 28, 2025 at 12:23:03 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=a \,{\mathrm e}^{\left (\mu +2 \lambda \right ) x} y^{2}+\left (b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}-\lambda \right ) y+c \,{\mathrm e}^{\mu x} \end{align*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 79
dsolve(diff(y(x),x)=a*exp((2*lambda+mu)*x)*y(x)^2+(b*exp((lambda+mu)*x)-lambda)*y(x)+c*exp(mu*x),y(x), singsol=all)
\[
y = \frac {{\mathrm e}^{-\lambda x} \left (\sqrt {4 a \,b^{2} c -b^{4}}\, \tan \left (\frac {\left (b \,{\mathrm e}^{x \left (\lambda +\mu \right )}+\left (\lambda +\mu \right ) c_{1} \right ) \sqrt {4 a \,b^{2} c -b^{4}}}{2 b^{2} \left (\lambda +\mu \right )}\right )-b^{2}\right )}{2 a b}
\]
✓ Solution by Mathematica
Time used: 3.004 (sec). Leaf size: 349
DSolve[D[y[x],x]==a*Exp[(2*\[Lambda]+\[Mu])*x]*y[x]^2+(b*Exp[(\[Lambda]+\[Mu])*x]-\[Lambda])*y[x]+c*Exp[\[Mu]*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {e^{\lambda (-x)} \left (b^2 e^{x (\lambda +\mu )} \left (\pi +i c_1 \left (e^{\sqrt {\frac {\left (b^2-4 a c\right ) e^{2 x (\lambda +\mu )}}{(\lambda +\mu )^2}}}-1\right )\right )-b (\lambda +\mu ) \sqrt {\frac {\left (b^2-4 a c\right ) e^{2 x (\lambda +\mu )}}{(\lambda +\mu )^2}} \left (\pi -i c_1 \left (e^{\sqrt {\frac {\left (b^2-4 a c\right ) e^{2 x (\lambda +\mu )}}{(\lambda +\mu )^2}}}+1\right )\right )-4 a c e^{x (\lambda +\mu )} \left (\pi +i c_1 \left (e^{\sqrt {\frac {\left (b^2-4 a c\right ) e^{2 x (\lambda +\mu )}}{(\lambda +\mu )^2}}}-1\right )\right )\right )}{2 a (\lambda +\mu ) \sqrt {\frac {\left (b^2-4 a c\right ) e^{2 x (\lambda +\mu )}}{(\lambda +\mu )^2}} \left (\pi -i c_1 \left (e^{\sqrt {\frac {\left (b^2-4 a c\right ) e^{2 x (\lambda +\mu )}}{(\lambda +\mu )^2}}}+1\right )\right )} \\
y(x)\to \frac {e^{\lambda (-x)} \left (-(\lambda +\mu ) e^{-x (\lambda +\mu )} \sqrt {\frac {\left (b^2-4 a c\right ) e^{2 x (\lambda +\mu )}}{(\lambda +\mu )^2}} \tanh \left (\frac {1}{2} \sqrt {\frac {\left (b^2-4 a c\right ) e^{2 x (\lambda +\mu )}}{(\lambda +\mu )^2}}\right )-b\right )}{2 a} \\
\end{align*}