30.1.19 problem 19

Internal problem ID [7409]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.2, Separable Equations. Exercises. page 46
Problem number : 19
Date solved : Tuesday, September 30, 2025 at 04:31:31 PM
CAS classification : [_separable]

\begin{align*} \frac {y^{\prime }}{2}&=\sqrt {1+y}\, \cos \left (x \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.489 (sec). Leaf size: 11

ode:=1/2*diff(y(x),x) = (1+y(x))^(1/2)*cos(x); 
ic:=[y(Pi) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \sin \left (x \right ) \left (\sin \left (x \right )+2\right ) \]

✓ Mathematica. Time used: 0.097 (sec). Leaf size: 69

ode=1/2*D[y[x],x]==Sqrt[1+y[x]]*Cos[x]; 
ic={y[Pi]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{4} \left (\int _{\pi }^x2 \cos (K[1])dK[1]-4\right ) \int _{\pi }^x2 \cos (K[1])dK[1]\\ y(x)&\to \frac {1}{4} \int _{\pi }^x2 \cos (K[1])dK[1] \left (\int _{\pi }^x2 \cos (K[1])dK[1]+4\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-sqrt(y(x) + 1)*cos(x) + Derivative(y(x), x)/2,0) 
ics = {y(pi): 0} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : Initial conditions produced too many solutions for constants