15.22.14 problem 14

Internal problem ID [3348]
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 : 14
Date solved : Monday, January 27, 2025 at 07:34:38 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+\frac {{y^{\prime }}^{2}}{2}-y&=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: 0.001 (sec). Leaf size: 20

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

\[ y \left (x \right ) = x -\frac {1}{4} x^{2}+\frac {1}{4} x^{3}-\frac {3}{32} x^{4}+\frac {1}{20} x^{5}-\frac {13}{480} x^{6}+\operatorname {O}\left (x^{7}\right ) \]

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 40

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

\[ y(x)\to -\frac {13 x^6}{480}+\frac {x^5}{20}-\frac {3 x^4}{32}+\frac {x^3}{4}-\frac {x^2}{4}+x \]