12.5.50 problem 49

Internal problem ID [1674]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Transformation of Nonlinear Equations into Separable Equations. Section 2.4 Page 68
Problem number : 49
Date solved : Monday, January 27, 2025 at 05:24:54 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`], _Riccati]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 16

dsolve(x*(ln(x))^2*diff(y(x),x)=-4*(ln(x))^2+y(x)*ln(x)+y(x)^2,y(x), singsol=all)
 

\[ y = 2 i \tan \left (2 i \ln \left (\ln \left (x \right )\right )+c_1 \right ) \ln \left (x \right ) \]

✓ Solution by Mathematica

Time used: 1.021 (sec). Leaf size: 64

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

\begin{align*} y(x)\to 2 i \log (x) \tan (2 i \log (\log (x))+c_1) \\ y(x)\to \frac {2 \log (x) \left (-\log ^4(x)+e^{2 i \text {Interval}[\{0,\pi \}]}\right )}{\log ^4(x)+e^{2 i \text {Interval}[\{0,\pi \}]}} \\ \end{align*}