60.1.40 problem 40

Internal problem ID [10054]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 40
Date solved : Tuesday, January 28, 2025 at 04:15:31 PM
CAS classification : [_Abel]

\begin{align*} y^{\prime }+3 a y^{3}+6 a x y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 57

dsolve(diff(y(x),x) + 3*a*y(x)^3 + 6*a*x*y(x)^2=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.303 (sec). Leaf size: 185

DSolve[D[y[x],x] + 3*a*y[x]^3 + 6*a*x*y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\frac {\sqrt [3]{-3} \sqrt [3]{a} x \operatorname {AiryAi}\left ((-3)^{2/3} a^{2/3} x^2-\frac {(-1)^{2/3}}{\sqrt [3]{3} \sqrt [3]{a} y(x)}\right )+\operatorname {AiryAiPrime}\left ((-3)^{2/3} a^{2/3} x^2-\frac {(-1)^{2/3}}{\sqrt [3]{3} \sqrt [3]{a} y(x)}\right )}{\sqrt [3]{-3} \sqrt [3]{a} x \operatorname {AiryBi}\left ((-3)^{2/3} a^{2/3} x^2-\frac {(-1)^{2/3}}{\sqrt [3]{3} \sqrt [3]{a} y(x)}\right )+\operatorname {AiryBiPrime}\left ((-3)^{2/3} a^{2/3} x^2-\frac {(-1)^{2/3}}{\sqrt [3]{3} \sqrt [3]{a} y(x)}\right )}+c_1=0,y(x)\right ] \]