60.1.257 problem 258

Internal problem ID [10271]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 258
Date solved : Monday, January 27, 2025 at 06:42:48 PM
CAS classification : [[_homogeneous, `class D`], _rational, _Bernoulli]

\begin{align*} 2 x^{2} y y^{\prime }+y^{2}-2 x^{3}-x^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.011 (sec). Leaf size: 33

dsolve(2*x^2*y(x)*diff(y(x),x)+y(x)^2-2*x^3-x^2=0,y(x), singsol=all)
 

\begin{align*} y &= \sqrt {{\mathrm e}^{\frac {1}{x}} c_{1} +x^{2}} \\ y &= -\sqrt {{\mathrm e}^{\frac {1}{x}} c_{1} +x^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.336 (sec). Leaf size: 195

DSolve[2*x^2*y[x]*D[y[x],x]+y[x]^2-2*x^3-x^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -e^{\left .\frac {1}{2}\right /x} \sqrt {2 \int _1^x\frac {1}{2} e^{-\frac {1}{K[1]}} (2 K[1]+1)dK[1]+c_1} \\ y(x)\to e^{\left .\frac {1}{2}\right /x} \sqrt {2 \int _1^x\frac {1}{2} e^{-\frac {1}{K[1]}} (2 K[1]+1)dK[1]+c_1} \\ y(x)\to -\sqrt {2} e^{\left .\frac {1}{2}\right /x} \sqrt {\int _1^x\frac {1}{2} e^{-\frac {1}{K[1]}} (2 K[1]+1)dK[1]} \\ y(x)\to \sqrt {2} e^{\left .\frac {1}{2}\right /x} \sqrt {\int _1^x\frac {1}{2} e^{-\frac {1}{K[1]}} (2 K[1]+1)dK[1]} \\ \end{align*}