54.1.281 problem 287

Internal problem ID [11595]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 287
Date solved : Tuesday, September 30, 2025 at 09:36:15 PM
CAS classification : [[_homogeneous, `class C`], _rational, _dAlembert]

\begin{align*} \left (2 y-4 x +1\right )^{2} y^{\prime }-\left (y-2 x \right )^{2}&=0 \end{align*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 56

ode:=(2*y(x)-4*x+1)^2*diff(y(x),x)-(y(x)-2*x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ -\frac {x}{7}+\frac {4 y}{7}-\frac {2 \ln \left (7 \left (y-2 x \right )^{2}+8 y-16 x +2\right )}{49}-\frac {9 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (7 y-14 x +4\right ) \sqrt {2}}{2}\right )}{98}-c_1 = 0 \]

✓ Mathematica. Time used: 0.137 (sec). Leaf size: 215

ode=(2*y[x]-4*x+1)^2*D[y[x],x]-(y[x]-2*x)^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {8 x-4 K[2]-1}{7 \left (28 x^2-28 K[2] x-16 x+7 K[2]^2+8 K[2]+2\right )}+\frac {1}{7} \left (4-7 \int _1^x\left (\frac {2 (8 K[1]-4 K[2]-1) (-28 K[1]+14 K[2]+8)}{7 \left (28 K[1]^2-28 K[2] K[1]-16 K[1]+7 K[2]^2+8 K[2]+2\right )^2}+\frac {8}{7 \left (28 K[1]^2-28 K[2] K[1]-16 K[1]+7 K[2]^2+8 K[2]+2\right )}\right )dK[1]\right )\right )dK[2]+\int _1^x\left (-\frac {2 (8 K[1]-4 y(x)-1)}{7 \left (28 K[1]^2-28 y(x) K[1]-16 K[1]+7 y(x)^2+8 y(x)+2\right )}-\frac {1}{7}\right )dK[1]=c_1,y(x)\right ] \]

✓ Sympy. Time used: 18.967 (sec). Leaf size: 70

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(-2*x + y(x))**2 + (-4*x + 2*y(x) + 1)**2*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} - \frac {x}{7} + \frac {4 y{\left (x \right )}}{7} - \frac {\left (8 + 9 \sqrt {2}\right ) \log {\left (2 x - y{\left (x \right )} - \frac {4}{7} - \frac {\sqrt {2}}{7} \right )}}{196} - \frac {\left (8 - 9 \sqrt {2}\right ) \log {\left (2 x - y{\left (x \right )} - \frac {4}{7} + \frac {\sqrt {2}}{7} \right )}}{196} = 0 \]