68.14.34 problem 34

Internal problem ID [17726]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.6, page 187
Problem number : 34
Date solved : Thursday, October 02, 2025 at 02:27:24 PM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} t y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }&=\frac {45}{8 t^{{7}/{2}}} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.047 (sec). Leaf size: 24

ode:=t*diff(diff(diff(diff(y(t),t),t),t),t)+2*diff(diff(diff(y(t),t),t),t) = 45/8/t^(7/2); 
ic:=[y(1) = 0, D(y)(1) = 0, (D@@2)(y)(1) = 1, (D@@3)(y)(1) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {13 t^{2}}{8}+\frac {2}{\sqrt {t}}-\frac {15 t \ln \left (t \right )}{4}+\frac {3 t}{2}-\frac {41}{8} \]

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 31

ode=t*D[y[t],{t,4}]+2*D[ y[t],{t,3}]==45/8*1/t^(7/2); 
ic={y[1]==0,Derivative[1][y][1]==0,Derivative[2][y][1]==1,Derivative[3][y][1]==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{8} \left (13 t^2+12 t+\frac {16}{\sqrt {t}}-30 t \log (t)-41\right ) \end{align*}

✓ Sympy. Time used: 0.179 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t*Derivative(y(t), (t, 4)) + 2*Derivative(y(t), (t, 3)) - 45/(8*t**(7/2)),0) 
ics = {y(1): 0, Subs(Derivative(y(t), t), t, 1): 0, Subs(Derivative(y(t), (t, 2)), t, 1): 1, Subs(Derivative(y(t), (t, 3)), t, 1): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {13 t^{2}}{8} - \frac {15 t \log {\left (t \right )}}{4} + \frac {3 t}{2} - \frac {41}{8} + \frac {2}{\sqrt {t}} \]