62.11.4 problem Ex 4

Internal problem ID [12846]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 18. Transformation of variables. Page 26
Problem number : Ex 4
Date solved : Tuesday, January 28, 2025 at 04:26:51 AM
CAS classification : [_rational, _Riccati]

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

✓ Solution by Maple

Time used: 0.051 (sec). Leaf size: 34

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

\[ y = \frac {\tanh \left (\frac {x^{a} \sqrt {b}\, \sqrt {c}+i c_{1} a}{a}\right ) \sqrt {c}\, x^{a}}{\sqrt {b}} \]

✓ Solution by Mathematica

Time used: 0.318 (sec). Leaf size: 153

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

\begin{align*} y(x)\to \frac {\sqrt {c} x^a \left (-\cos \left (\frac {\sqrt {-b} \sqrt {c} x^a}{a}\right )+c_1 \sin \left (\frac {\sqrt {-b} \sqrt {c} x^a}{a}\right )\right )}{\sqrt {-b} \left (\sin \left (\frac {\sqrt {-b} \sqrt {c} x^a}{a}\right )+c_1 \cos \left (\frac {\sqrt {-b} \sqrt {c} x^a}{a}\right )\right )} \\ y(x)\to \frac {\sqrt {c} x^a \tan \left (\frac {\sqrt {-b} \sqrt {c} x^a}{a}\right )}{\sqrt {-b}} \\ \end{align*}