15.22.8 problem 8

Internal problem ID [3342]
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 : 8
Date solved : Monday, January 27, 2025 at 07:34:33 AM
CAS classification : [`y=_G(x,y')`]

\begin{align*} y^{\prime }&=\sqrt {1+y x} \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 16

Order:=4; 
dsolve([diff(y(x),x)=sqrt(1+x*y(x)),y(0) = 1],y(x),type='series',x=0);
 

\[ y \left (x \right ) = 1+x +\frac {1}{4} x^{2}+\frac {1}{8} x^{3}+\operatorname {O}\left (x^{4}\right ) \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 20

AsymptoticDSolveValue[{D[y[x],x]==Sqrt[1+x*y[x]],{y[0]==1}},y[x],{x,0,"4"-1}]
 

\[ y(x)\to \frac {x^3}{8}+\frac {x^2}{4}+x+1 \]