47.2.44 problem Example 5

Internal problem ID [7460]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.2 Homogeneous equations problems. page 12
Problem number : Example 5
Date solved : Monday, January 27, 2025 at 03:01:03 PM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

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

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 67

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

\begin{align*} y &= \frac {1}{\sqrt {\sqrt {2 \ln \left (x \right )+c_{1}}\, x}} \\ y &= \frac {1}{\sqrt {-\sqrt {2 \ln \left (x \right )+c_{1}}\, x}} \\ y &= -\frac {1}{\sqrt {\sqrt {2 \ln \left (x \right )+c_{1}}\, x}} \\ y &= -\frac {1}{\sqrt {-\sqrt {2 \ln \left (x \right )+c_{1}}\, x}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.614 (sec). Leaf size: 92

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

\begin{align*} y(x)\to -\frac {1}{\sqrt [4]{x^2 (2 \log (x)+c_1)}} \\ y(x)\to -\frac {i}{\sqrt [4]{x^2 (2 \log (x)+c_1)}} \\ y(x)\to \frac {i}{\sqrt [4]{x^2 (2 \log (x)+c_1)}} \\ y(x)\to \frac {1}{\sqrt [4]{x^2 (2 \log (x)+c_1)}} \\ y(x)\to 0 \\ \end{align*}