74.7.38 problem 38

Internal problem ID [16115]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.5, page 64
Problem number : 38
Date solved : Tuesday, January 28, 2025 at 08:42:06 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

With initial conditions

\begin{align*} y \left (1\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 2.307 (sec). Leaf size: 22

dsolve([(t^3+y(t)^2*sqrt(t^2+y(t)^2))-(t*y(t)*sqrt(t^2+y(t)^2))*diff(y(t),t)=0,y(1) = 1],y(t), singsol=all)
 

\[ y = \sqrt {-1+\left (3 \ln \left (t \right )+2 \sqrt {2}\right )^{{2}/{3}}}\, t \]

✓ Solution by Mathematica

Time used: 20.482 (sec). Leaf size: 80

DSolve[{(t^3+y[t]^2*Sqrt[t^2+y[t]^2])-(t*y[t]*Sqrt[t^2+y[t]^2])*D[y[t],t]==0,{y[1]==1}},y[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} y(t)\to \sqrt {\sqrt [3]{-t^6 \left (-9 \log ^2(t)+12 \sqrt {2} \log (t)-8\right )}-t^2} \\ y(t)\to \sqrt {\sqrt [3]{t^6 \left (9 \log ^2(t)+12 \sqrt {2} \log (t)+8\right )}-t^2} \\ \end{align*}