61.2.24 problem 24
Internal
problem
ID
[12030]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
1.2.2.
Equations
Containing
Power
Functions
Problem
number
:
24
Date
solved
:
Monday, January 27, 2025 at 11:51:45 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=a y^{2}+b y+c x +k \end{align*}
✓ Solution by Maple
Time used: 0.020 (sec). Leaf size: 194
dsolve(diff(y(x),x)=a*y(x)^2+b*y(x)+c*x+k,y(x), singsol=all)
\[
y = \frac {2 \left (\frac {c}{\sqrt {a}}\right )^{{1}/{3}} \sqrt {a}\, \left (\operatorname {AiryAi}\left (1, -\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{{2}/{3}} a}\right ) c_{1} +\operatorname {AiryBi}\left (1, -\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{{2}/{3}} a}\right )\right )-\left (c_{1} \operatorname {AiryAi}\left (-\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{{2}/{3}} a}\right )+\operatorname {AiryBi}\left (-\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{{2}/{3}} a}\right )\right ) b}{2 a \left (c_{1} \operatorname {AiryAi}\left (-\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{{2}/{3}} a}\right )+\operatorname {AiryBi}\left (-\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{{2}/{3}} a}\right )\right )}
\]
✓ Solution by Mathematica
Time used: 0.271 (sec). Leaf size: 284
DSolve[D[y[x],x]==a*y[x]^2+b*y[x]+c*x+k,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {c \left (-b (-a c)^{2/3} \operatorname {AiryBi}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )+2 a c \operatorname {AiryBiPrime}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )+c_1 \left (2 a c \operatorname {AiryAiPrime}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )-b (-a c)^{2/3} \operatorname {AiryAi}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )\right )\right )}{2 (-a c)^{5/3} \left (\operatorname {AiryBi}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )+c_1 \operatorname {AiryAi}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )\right )} \\
y(x)\to -\frac {\frac {2 \sqrt [3]{-a c} \operatorname {AiryAiPrime}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )}{\operatorname {AiryAi}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )}+b}{2 a} \\
\end{align*}