83.34.5 problem 5

Internal problem ID [19418]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Exercise VII (H) at page 118
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 08:32:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x^{4} y^{\prime \prime }&=\left (x^{3}+2 y x \right ) y^{\prime }-4 y^{2} \end{align*}

✓ Solution by Maple

Time used: 0.032 (sec). Leaf size: 22

dsolve(x^4*diff(y(x),x$2)=(x^3+2*x*y(x))*diff(y(x),x)-4*y(x)^2,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 70.828 (sec). Leaf size: 83

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

\[ y(x)\to \frac {x^2 \left (\left (1-i \sqrt {-1-c_1}\right ) x^{2 i \sqrt {-1-c_1}}+\left (1+i \sqrt {-1-c_1}\right ) c_2\right )}{x^{2 i \sqrt {-1-c_1}}+c_2} \]