86.3.14 problem 12

Internal problem ID [23105]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 4. Linear equations of the first order. Exercise 4a at page 56
Problem number : 12
Date solved : Thursday, October 02, 2025 at 09:21:59 PM
CAS classification : [_linear]

\begin{align*} \left (x^{2}+1\right ) y^{\prime }+y x&=\left (x^{2}+1\right )^{{3}/{2}} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=7 \\ \end{align*}

✓ Maple. Time used: 0.017 (sec). Leaf size: 21

ode:=(x^2+1)*diff(y(x),x)+x*y(x) = (x^2+1)^(3/2); 
ic:=[y(0) = 7]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {x^{3}+3 x +21}{3 \sqrt {x^{2}+1}} \]

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 26

ode=(1+x^2)*D[y[x],x]+x*y[x]==(1+x^2)^(3/2); 
ic={y[0]==7}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x^3+3 x+21}{3 \sqrt {x^2+1}} \end{align*}

✓ Sympy. Time used: 0.304 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x) - (x**2 + 1)**(3/2) + (x**2 + 1)*Derivative(y(x), x),0) 
ics = {y(0): 7} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\frac {x^{3}}{3} + x + 7}{\sqrt {x^{2} + 1}} \]