61.3.8 problem 8

Internal problem ID [12092]
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
Date solved : Tuesday, January 28, 2025 at 12:21:51 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}+a \,{\mathrm e}^{8 \lambda x}+b \,{\mathrm e}^{6 \lambda x}+c \,{\mathrm e}^{4 \lambda x}-\lambda ^{2} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 1020

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: 2.617 (sec). Leaf size: 1515

DSolve[D[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],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {-e^{2 x \lambda } \operatorname {Hypergeometric1F1}\left (\frac {-i b^2+4 i a c+40 a^{3/2} \lambda }{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 ) b^3-2 a e^{4 x \lambda } \operatorname {Hypergeometric1F1}\left (\frac {-i b^2+4 i a c+40 a^{3/2} \lambda }{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 ) b^2+8 i a^{3/2} e^{2 x \lambda } \lambda \operatorname {Hypergeometric1F1}\left (\frac {-i b^2+4 i a c+8 a^{3/2} \lambda }{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+4 a c e^{2 x \lambda } \operatorname {Hypergeometric1F1}\left (\frac {-i b^2+4 i a c+40 a^{3/2} \lambda }{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 ) b-8 i a^{3/2} e^{2 x \lambda } \lambda \operatorname {Hypergeometric1F1}\left (\frac {-i b^2+4 i a c+40 a^{3/2} \lambda }{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 ) b+8 a^{3/2} \lambda \left (2 i e^{4 x \lambda } a+2 \lambda \sqrt {a}+i b e^{2 x \lambda }\right ) c_1 \operatorname {HermiteH}\left (\frac {i \left (b^2-4 a c+8 i a^{3/2} \lambda \right )}{16 a^{3/2} \lambda },\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (2 e^{2 x \lambda } a+b\right )}{a^{3/4} \sqrt {\lambda }}\right )+2 \sqrt [4]{-1} \sqrt {2} a^{3/4} e^{2 x \lambda } \sqrt {\lambda } \left (-i b^2+4 i a c+8 a^{3/2} \lambda \right ) c_1 \operatorname {HermiteH}\left (\frac {i \left (b^2-4 a c+24 i a^{3/2} \lambda \right )}{16 a^{3/2} \lambda },\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (2 e^{2 x \lambda } a+b\right )}{a^{3/4} \sqrt {\lambda }}\right )+16 a^2 \lambda ^2 \operatorname {Hypergeometric1F1}\left (\frac {-i b^2+4 i a c+8 a^{3/2} \lambda }{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 )+16 i a^{5/2} e^{4 x \lambda } \lambda \operatorname {Hypergeometric1F1}\left (\frac {-i b^2+4 i a c+8 a^{3/2} \lambda }{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 )+8 a^2 c e^{4 x \lambda } \operatorname {Hypergeometric1F1}\left (\frac {-i b^2+4 i a c+40 a^{3/2} \lambda }{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 i a^{5/2} e^{4 x \lambda } \lambda \operatorname {Hypergeometric1F1}\left (\frac {-i b^2+4 i a c+40 a^{3/2} \lambda }{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+8 i a^{3/2} \lambda \right )}{16 a^{3/2} \lambda },\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 {-i b^2+4 i a c+8 a^{3/2} \lambda }{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 {\frac {\sqrt [4]{-1} \sqrt {2} e^{2 \lambda x} \left (8 a^{3/2} \lambda +4 i a c-i b^2\right ) \operatorname {HermiteH}\left (\frac {i \left (b^2-4 a c+24 i a^{3/2} \lambda \right )}{16 a^{3/2} \lambda },\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (2 e^{2 x \lambda } a+b\right )}{a^{3/4} \sqrt {\lambda }}\right )}{\sqrt {\lambda } \operatorname {HermiteH}\left (\frac {i \left (b^2-4 a c+8 i a^{3/2} \lambda \right )}{16 a^{3/2} \lambda },\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (2 e^{2 x \lambda } a+b\right )}{a^{3/4} \sqrt {\lambda }}\right )}+4 a^{3/4} \left (2 \sqrt {a} \lambda +2 i a e^{4 \lambda x}+i b e^{2 \lambda x}\right )}{8 a^{5/4}} \\ y(x)\to \frac {\frac {\sqrt [4]{-1} \sqrt {2} e^{2 \lambda x} \left (8 a^{3/2} \lambda +4 i a c-i b^2\right ) \operatorname {HermiteH}\left (\frac {i \left (b^2-4 a c+24 i a^{3/2} \lambda \right )}{16 a^{3/2} \lambda },\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (2 e^{2 x \lambda } a+b\right )}{a^{3/4} \sqrt {\lambda }}\right )}{\sqrt {\lambda } \operatorname {HermiteH}\left (\frac {i \left (b^2-4 a c+8 i a^{3/2} \lambda \right )}{16 a^{3/2} \lambda },\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (2 e^{2 x \lambda } a+b\right )}{a^{3/4} \sqrt {\lambda }}\right )}+4 a^{3/4} \left (2 \sqrt {a} \lambda +2 i a e^{4 \lambda x}+i b e^{2 \lambda x}\right )}{8 a^{5/4}} \\ \end{align*}