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 : Monday, January 27, 2025 at 04:09:13 PM
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*}

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

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

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

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