60.1.179 problem 180

Internal problem ID [10193]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 180
Date solved : Monday, January 27, 2025 at 06:37:26 PM
CAS classification : [[_homogeneous, `class D`], _rational, _Riccati]

\begin{align*} \left (a \,x^{2}+b x +c \right ) \left (-y+x y^{\prime }\right )-y^{2}+x^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 58

dsolve((a*x^2+b*x+c)*(x*diff(y(x),x)-y(x)) - y(x)^2 + x^2=0,y(x), singsol=all)
 

\[ y = -\tanh \left (\frac {c_{1} \sqrt {4 a c -b^{2}}+2 \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\right ) x \]

✓ Solution by Mathematica

Time used: 0.247 (sec). Leaf size: 53

DSolve[(a*x^2+b*x+c)*(x*D[y[x],x]-y[x]) - y[x]^2 + x^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {1}{(K[1]-1) (K[1]+1)}dK[1]=\int _1^x\frac {1}{a K[2]^2+b K[2]+c}dK[2]+c_1,y(x)\right ] \]