75.8.9 problem 207

Internal problem ID [16754]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 8. First order not solved for the derivative. Exercises page 67
Problem number : 207
Date solved : Monday, March 31, 2025 at 03:16:01 PM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} {y^{\prime }}^{2}-4 x y^{\prime }+2 y+2 x^{2}&=0 \end{align*}

✓ Maple. Time used: 0.043 (sec). Leaf size: 77

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

\begin{align*} y &= x^{2} \\ y &= \frac {1}{2} x^{2}+c_1 x -\frac {1}{2} c_1^{2} \\ y &= \frac {1}{2} x^{2}-c_1 x -\frac {1}{2} c_1^{2} \\ y &= \frac {1}{2} x^{2}-c_1 x -\frac {1}{2} c_1^{2} \\ y &= \frac {1}{2} x^{2}+c_1 x -\frac {1}{2} c_1^{2} \\ \end{align*}

✓ Mathematica. Time used: 14.58 (sec). Leaf size: 502

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

\begin{align*} \text {Solve}\left [\frac {-\frac {\left (-y(x)+\sqrt {2} \sqrt {y(x)+1}-1\right ) \text {arctanh}\left (\frac {y(x)+x \left (\sqrt {2} \sqrt {y(x)+1}-1\right )}{\sqrt {x^2-y(x)} \sqrt {y(x)-2 \sqrt {2} \sqrt {y(x)+1}+3}}\right )}{\sqrt {y(x)-2 \sqrt {2} \sqrt {y(x)+1}+3}}-\frac {\left (y(x)+\sqrt {2} \sqrt {y(x)+1}+1\right ) \text {arctanh}\left (\frac {y(x)-x \left (\sqrt {2} \sqrt {y(x)+1}+1\right )}{\sqrt {x^2-y(x)} \sqrt {y(x)+2 \sqrt {2} \sqrt {y(x)+1}+3}}\right )}{\sqrt {y(x)+2 \sqrt {2} \sqrt {y(x)+1}+3}}+\sqrt {y(x)+1} \log \left (-x^2+2 y(x)+2 x+1\right )}{2 \sqrt {y(x)+1}}&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {\frac {\left (-y(x)+\sqrt {2} \sqrt {y(x)+1}-1\right ) \text {arctanh}\left (\frac {y(x)+x \left (\sqrt {2} \sqrt {y(x)+1}-1\right )}{\sqrt {x^2-y(x)} \sqrt {y(x)-2 \sqrt {2} \sqrt {y(x)+1}+3}}\right )}{\sqrt {y(x)-2 \sqrt {2} \sqrt {y(x)+1}+3}}+\frac {\left (y(x)+\sqrt {2} \sqrt {y(x)+1}+1\right ) \text {arctanh}\left (\frac {y(x)-x \left (\sqrt {2} \sqrt {y(x)+1}+1\right )}{\sqrt {x^2-y(x)} \sqrt {y(x)+2 \sqrt {2} \sqrt {y(x)+1}+3}}\right )}{\sqrt {y(x)+2 \sqrt {2} \sqrt {y(x)+1}+3}}+\sqrt {y(x)+1} \log \left (-x^2+2 y(x)+2 x+1\right )}{2 \sqrt {y(x)+1}}&=c_1,y(x)\right ] \\ y(x)\to x^2 \\ \end{align*}

✓ Sympy. Time used: 2.936 (sec). Leaf size: 12

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

\[ y{\left (x \right )} = x^{2} - \frac {\left (C_{1} + x\right )^{2}}{2} \]