15.22.12 problem 12

Internal problem ID [3346]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 40, page 186
Problem number : 12
Date solved : Monday, January 27, 2025 at 07:34:37 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

Using series method with expansion around

\begin{align*} 0 \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.000 (sec). Leaf size: 14

Order:=7; 
dsolve([diff(y(x),x$2)+2*y(x)*diff(y(x),x)=0,y(0) = 0, D(y)(0) = 1],y(x),type='series',x=0);
 

\[ y \left (x \right ) = x -\frac {1}{3} x^{3}+\frac {2}{15} x^{5}+\operatorname {O}\left (x^{7}\right ) \]

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 19

AsymptoticDSolveValue[{D[y[x],{x,2}]+2*y[x]*D[y[x],x]==0,{y[0]==0,Derivative[1][y][0] ==1}},y[x],{x,0,"7"-1}]
 

\[ y(x)\to \frac {2 x^5}{15}-\frac {x^3}{3}+x \]