29.3.2 problem 56

Internal problem ID [4664]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 3
Problem number : 56
Date solved : Monday, January 27, 2025 at 09:30:43 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=a +b x +c y^{2} \end{align*}

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 85

dsolve(diff(y(x),x) = a+b*x+c*y(x)^2,y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {\left (\frac {b}{\sqrt {c}}\right )^{{1}/{3}} \left (\operatorname {AiryAi}\left (1, -\frac {b x +a}{\left (\frac {b}{\sqrt {c}}\right )^{{2}/{3}}}\right ) c_{1} +\operatorname {AiryBi}\left (1, -\frac {b x +a}{\left (\frac {b}{\sqrt {c}}\right )^{{2}/{3}}}\right )\right )}{\sqrt {c}\, \left (c_{1} \operatorname {AiryAi}\left (-\frac {b x +a}{\left (\frac {b}{\sqrt {c}}\right )^{{2}/{3}}}\right )+\operatorname {AiryBi}\left (-\frac {b x +a}{\left (\frac {b}{\sqrt {c}}\right )^{{2}/{3}}}\right )\right )} \]

✓ Solution by Mathematica

Time used: 0.203 (sec). Leaf size: 143

DSolve[D[y[x],x]==a+b x+c y[x]^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {b \left (\operatorname {AiryBiPrime}\left (-\frac {c (a+b x)}{(-b c)^{2/3}}\right )+c_1 \operatorname {AiryAiPrime}\left (-\frac {c (a+b x)}{(-b c)^{2/3}}\right )\right )}{(-b c)^{2/3} \left (\operatorname {AiryBi}\left (-\frac {c (a+b x)}{(-b c)^{2/3}}\right )+c_1 \operatorname {AiryAi}\left (-\frac {c (a+b x)}{(-b c)^{2/3}}\right )\right )} \\ y(x)\to \frac {b \operatorname {AiryAiPrime}\left (-\frac {c (a+b x)}{(-b c)^{2/3}}\right )}{(-b c)^{2/3} \operatorname {AiryAi}\left (-\frac {c (a+b x)}{(-b c)^{2/3}}\right )} \\ \end{align*}