Internal problem ID [9669]
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: 8.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}-a \,{\mathrm e}^{8 x \lambda }-b \,{\mathrm e}^{6 x \lambda }-c \,{\mathrm e}^{4 x \lambda }+\lambda ^{2}=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 2495
dsolve(diff(y(x),x)=y(x)^2+a*exp(8*lambda*x)+b*exp(6*lambda*x)+c*exp(4*lambda*x)-lambda^2,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 2.408 (sec). Leaf size: 1239
DSolve[y'[x]==y[x]^2+a*Exp[8*\[Lambda]*x]+b*Exp[6*\[Lambda]*x]+c*Exp[4*\[Lambda]*x]-\[Lambda]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {8 \lambda \left (-2 i e^{4 x \lambda } a+2 \lambda \sqrt {a}-i b e^{2 x \lambda }\right ) \operatorname {Hypergeometric1F1}\left (\frac {1}{4}-\frac {i \left (b^2-4 a c\right )}{32 a^{3/2} \lambda },\frac {1}{2},\frac {i \left (2 e^{2 x \lambda } a+b\right )^2}{8 a^{3/2} \lambda }\right ) a^{3/2}+\frac {(1+i) 2^{\frac {i \left (b^2-4 a c\right )}{16 a^{3/2} \lambda }+4} \sqrt {\pi } \lambda c_1 \left (\frac {(4-4 i) \sqrt {2} a^{3/2} \lambda \left (2 e^{4 x \lambda } a+2 i \lambda \sqrt {a}+b e^{2 x \lambda }\right ) \operatorname {Hypergeometric1F1}\left (\frac {1}{4}-\frac {i \left (b^2-4 a c\right )}{32 a^{3/2} \lambda },\frac {1}{2},\frac {i \left (2 e^{2 x \lambda } a+b\right )^2}{8 a^{3/2} \lambda }\right )}{b^2-4 a c-8 i a^{3/2} \lambda }-\sqrt [4]{-1} e^{2 x \lambda } \left (2 e^{2 x \lambda } a+b\right ) \operatorname {Hypergeometric1F1}\left (\frac {1}{4}-\frac {i \left (b^2-4 a c\right )}{32 a^{3/2} \lambda },\frac {3}{2},\frac {i \left (2 e^{2 x \lambda } a+b\right )^2}{8 a^{3/2} \lambda }\right )\right ) a^{3/2}}{\operatorname {Gamma}\left (-\frac {i \left (b^2-4 a c\right )}{32 a^{3/2} \lambda }-\frac {1}{4}\right )}+\frac {(1+i) 2^{\frac {i \left (b^2-4 a c\right )}{16 a^{3/2} \lambda }+1} \sqrt {\pi } \sqrt {\lambda } c_1 \left ((8-8 i) \sqrt [4]{-1} a^{3/2} e^{2 x \lambda } \lambda \operatorname {Hypergeometric1F1}\left (\frac {3}{4}-\frac {i \left (b^2-4 a c\right )}{32 a^{3/2} \lambda },\frac {1}{2},\frac {i \left (2 e^{2 x \lambda } a+b\right )^2}{8 a^{3/2} \lambda }\right )-i \sqrt {2} \left (2 e^{2 x \lambda } a+b\right ) \left (2 e^{4 x \lambda } a-2 i \lambda \sqrt {a}+b e^{2 x \lambda }\right ) \operatorname {Hypergeometric1F1}\left (\frac {3}{4}-\frac {i \left (b^2-4 a c\right )}{32 a^{3/2} \lambda },\frac {3}{2},\frac {i \left (2 e^{2 x \lambda } a+b\right )^2}{8 a^{3/2} \lambda }\right )\right ) a^{3/4}}{\operatorname {Gamma}\left (\frac {1}{4}-\frac {i \left (b^2-4 a c\right )}{32 a^{3/2} \lambda }\right )}-e^{2 x \lambda } \left (2 e^{2 x \lambda } a+b\right ) \left (b^2-4 a c-8 i a^{3/2} \lambda \right ) \operatorname {Hypergeometric1F1}\left (\frac {1}{4}-\frac {i \left (b^2-4 a c\right )}{32 a^{3/2} \lambda },\frac {3}{2},\frac {i \left (2 e^{2 x \lambda } a+b\right )^2}{8 a^{3/2} \lambda }\right )}{16 a^2 \lambda \left (c_1 \operatorname {HermiteH}\left (\frac {i \left (b^2-4 a c\right )}{16 a^{3/2} \lambda }-\frac {1}{2},\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (2 e^{2 x \lambda } a+b\right )}{a^{3/4} \sqrt {\lambda }}\right )+\operatorname {Hypergeometric1F1}\left (\frac {1}{4}-\frac {i \left (b^2-4 a c\right )}{32 a^{3/2} \lambda },\frac {1}{2},\frac {i \left (2 e^{2 x \lambda } a+b\right )^2}{8 a^{3/2} \lambda }\right )\right )} \\ y(x)\to \frac {\left (\frac {1}{8}-\frac {i}{8}\right ) e^{2 \lambda x} \left (8 i a^{3/2} \lambda -4 a c+b^2\right ) \operatorname {HermiteH}\left (\frac {i \left (b^2-4 a c\right )}{16 a^{3/2} \lambda }-\frac {3}{2},\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (2 e^{2 x \lambda } a+b\right )}{a^{3/4} \sqrt {\lambda }}\right )}{a^{5/4} \sqrt {\lambda } \operatorname {HermiteH}\left (\frac {i \left (b^2-4 a c\right )}{16 a^{3/2} \lambda }-\frac {1}{2},\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (2 e^{2 x \lambda } a+b\right )}{a^{3/4} \sqrt {\lambda }}\right )}+\frac {i e^{2 \lambda x} \left (2 a e^{2 \lambda x}+b\right )}{2 \sqrt {a}}+\lambda \\ y(x)\to \frac {\left (\frac {1}{8}-\frac {i}{8}\right ) e^{2 \lambda x} \left (8 i a^{3/2} \lambda -4 a c+b^2\right ) \operatorname {HermiteH}\left (\frac {i \left (b^2-4 a c\right )}{16 a^{3/2} \lambda }-\frac {3}{2},\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (2 e^{2 x \lambda } a+b\right )}{a^{3/4} \sqrt {\lambda }}\right )}{a^{5/4} \sqrt {\lambda } \operatorname {HermiteH}\left (\frac {i \left (b^2-4 a c\right )}{16 a^{3/2} \lambda }-\frac {1}{2},\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (2 e^{2 x \lambda } a+b\right )}{a^{3/4} \sqrt {\lambda }}\right )}+\frac {i e^{2 \lambda x} \left (2 a e^{2 \lambda x}+b\right )}{2 \sqrt {a}}+\lambda \\ \end{align*}