53.4.34 problem 37

Internal problem ID [8522]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 101. Independent variable missing. EXERCISES Page 324
Problem number : 37
Date solved : Wednesday, March 05, 2025 at 06:02:21 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} x^{4} y^{\prime \prime }&=y^{\prime } \left (y^{\prime }+x^{3}\right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.086 (sec). Leaf size: 24

ode:=x^4*diff(diff(y(x),x),x) = diff(y(x),x)*(diff(y(x),x)+x^3); 
ic:=y(1) = 2, D(y)(1) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = x^{2}-\ln \left (-x^{2}-1\right )+1+\ln \left (2\right )+i \pi \]

✓ Mathematica. Time used: 0.893 (sec). Leaf size: 20

ode=x^4*D[y[x],{x,2}]==D[y[x],x]*(D[y[x],x]+x^3); 
ic={y[1]==2,Derivative[1][y][1]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to x^2-\log \left (x^2+1\right )+1+\log (2) \]

✓ Sympy. Time used: 0.957 (sec). Leaf size: 17

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

\[ y{\left (x \right )} = x^{2} - \log {\left (x^{2} + 1 \right )} + \log {\left (2 \right )} + 1 \]