15.18.40 problem 40

Internal problem ID [3283]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 35, page 157
Problem number : 40
Date solved : Monday, January 27, 2025 at 07:30:50 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y y^{\prime \prime }&=2 {y^{\prime }}^{2}+y^{2} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=\sqrt {3} \end{align*}

✓ Solution by Maple

Time used: 0.076 (sec). Leaf size: 16

dsolve([y(x)*diff(y(x),x$2)=2*diff(y(x),x)^2+y(x)^2,y(0) = 1, D(y)(0) = 3^(1/2)],y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {1}{-\sin \left (x \right ) \sqrt {3}+\cos \left (x \right )} \]

✓ Solution by Mathematica

Time used: 0.201 (sec). Leaf size: 19

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

\[ y(x)\to \frac {1}{2} \csc \left (\frac {1}{6} (\pi -6 x)\right ) \]