15.2.23 problem 23

Internal problem ID [2893]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 6, page 25
Problem number : 23
Date solved : Monday, January 27, 2025 at 06:27:34 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} y^{\prime }&=\frac {y}{x}+\tanh \left (\frac {y}{x}\right ) \end{align*}

✓ Solution by Maple

Time used: 0.412 (sec). Leaf size: 103

dsolve(diff(y(x),x)=y(x)/x+tanh(y(x)/x),y(x), singsol=all)
 

\begin{align*} y &= \operatorname {arctanh}\left (\frac {-c_1 \,x^{2}+\sqrt {c_1 \,x^{2} \left (c_1 \,x^{2}-1\right )}}{-c_1 \,x^{2}+\sqrt {c_1 \,x^{2} \left (c_1 \,x^{2}-1\right )}+1}\right ) x \\ y &= \operatorname {arctanh}\left (\frac {c_1 \,x^{2}+\sqrt {c_1 \,x^{2} \left (c_1 \,x^{2}-1\right )}}{c_1 \,x^{2}-1+\sqrt {c_1 \,x^{2} \left (c_1 \,x^{2}-1\right )}}\right ) x \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.974 (sec). Leaf size: 19

DSolve[D[y[x],x]==y[x]/x+Tanh[y[x]/x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to x \text {arcsinh}\left (e^{c_1} x\right ) \\ y(x)\to 0 \\ \end{align*}