14.1.2 problem 2

Internal problem ID [2473]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 1. First order differential equations. Section 1.2. Linear equations. Excercises page 9
Problem number : 2
Date solved : Sunday, March 30, 2025 at 12:02:28 AM
CAS classification : [_separable]

\begin{align*} y^{\prime }+y \sqrt {t}\, \sin \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 33

ode:=diff(y(t),t)+y(t)*t^(1/2)*sin(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = c_1 \,{\mathrm e}^{\sqrt {t}\, \cos \left (t \right )-\frac {\sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {t}}{\sqrt {\pi }}\right )}{2}} \]

✓ Mathematica. Time used: 0.056 (sec). Leaf size: 66

ode=D[y[t],t]+y[t]*Sqrt[t]*Sin[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)\to c_1 \exp \left (\frac {i \left (\sqrt {-i t} \Gamma \left (\frac {3}{2},-i t\right )-\sqrt {i t} \Gamma \left (\frac {3}{2},i t\right )\right )}{2 \sqrt {t}}\right ) \\ y(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 1.113 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(sqrt(t)*y(t)*sin(t) + Derivative(y(t), t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{1} e^{\sqrt {t} \cos {\left (t \right )} - \frac {\sqrt {2} \sqrt {\pi } C\left (\frac {\sqrt {2} \sqrt {t}}{\sqrt {\pi }}\right )}{2}} \]