61.3.21 problem 21
Internal
problem
ID
[12105]
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
:
21
Date
solved
:
Tuesday, January 28, 2025 at 12:24:28 AM
CAS
classification
:
[_Riccati]
\begin{align*} \left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+c \right ) \left (y^{\prime }-y^{2}\right )+a \,\lambda ^{2} {\mathrm e}^{\lambda x}+b \,\mu ^{2} {\mathrm e}^{\mu x}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 176
dsolve((a*exp(lambda*x)+b*exp(mu*x)+c)*(diff(y(x),x)-y(x)^2)+a*lambda^2*exp(lambda*x)+b*mu^2*exp(mu*x)=0,y(x), singsol=all)
\[
y = \frac {\left (-\left (\lambda +\mu \right ) {\mathrm e}^{x \left (\lambda +\mu \right )} a b -a^{2} \lambda \,{\mathrm e}^{2 \lambda x}-{\mathrm e}^{2 \mu x} b^{2} \mu -c \left (a \lambda \,{\mathrm e}^{\lambda x}+b \mu \,{\mathrm e}^{\mu x}\right )\right ) \left (\int \frac {1}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}d x \right )-a b c_{1} \left (\lambda +\mu \right ) {\mathrm e}^{x \left (\lambda +\mu \right )}-{\mathrm e}^{2 \lambda x} c_{1} a^{2} \lambda -{\mathrm e}^{\lambda x} c_{1} a c \lambda -{\mathrm e}^{2 \mu x} c_{1} b^{2} \mu -{\mathrm e}^{\mu x} c_{1} b c \mu -1}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2} \left (c_{1} +\int \frac {1}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}d x \right )}
\]
✓ Solution by Mathematica
Time used: 6.357 (sec). Leaf size: 393
DSolve[(a*Exp[\[Lambda]*x]+b*Exp[\[Mu]*x]+c)*(D[y[x],x]-y[x]^2)+a*\[Lambda]^2*Exp[\[Lambda]*x]+b*\[Mu]^2*Exp[\[Mu]*x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^x-\frac {-a e^{\lambda K[1]} \lambda ^2-b e^{\mu K[1]} \mu ^2+a e^{\lambda K[1]} y(x)^2+b e^{\mu K[1]} y(x)^2+c y(x)^2}{\left (e^{\lambda K[1]} a+b e^{\mu K[1]}+c\right ) \left (a e^{\lambda K[1]} \lambda +b e^{\mu K[1]} \mu +a e^{\lambda K[1]} y(x)+b e^{\mu K[1]} y(x)+c y(x)\right )^2}dK[1]+\int _1^{y(x)}\left (\frac {1}{\left (a e^{x \lambda } \lambda +b e^{x \mu } \mu +a e^{x \lambda } K[2]+b e^{x \mu } K[2]+c K[2]\right )^2}-\int _1^x\left (\frac {2 \left (-a e^{\lambda K[1]} \lambda ^2-b e^{\mu K[1]} \mu ^2+a e^{\lambda K[1]} K[2]^2+b e^{\mu K[1]} K[2]^2+c K[2]^2\right )}{\left (a e^{\lambda K[1]} \lambda +b e^{\mu K[1]} \mu +a e^{\lambda K[1]} K[2]+b e^{\mu K[1]} K[2]+c K[2]\right )^3}-\frac {2 a e^{\lambda K[1]} K[2]+2 b e^{\mu K[1]} K[2]+2 c K[2]}{\left (e^{\lambda K[1]} a+b e^{\mu K[1]}+c\right ) \left (a e^{\lambda K[1]} \lambda +b e^{\mu K[1]} \mu +a e^{\lambda K[1]} K[2]+b e^{\mu K[1]} K[2]+c K[2]\right )^2}\right )dK[1]\right )dK[2]=c_1,y(x)\right ]
\]