83.22.9 problem 9

Internal problem ID [19263]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (E) at page 63
Problem number : 9
Date solved : Tuesday, January 28, 2025 at 01:17:36 PM
CAS classification : [_quadrature]

\begin{align*} 2 {y^{\prime }}^{3}-\left (2 x +4 \sin \left (x \right )-\cos \left (x \right )\right ) {y^{\prime }}^{2}-\left (x \cos \left (x \right )-4 x \sin \left (x \right )+\sin \left (2 x \right )\right ) y^{\prime }+x \sin \left (2 x \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.030 (sec). Leaf size: 29

dsolve(2*diff(y(x),x)^3-(2*x+4*sin(x)-cos(x))*diff(y(x),x)^2-(x*cos(x)-4*x*sin(x)+sin(2*x))*diff(y(x),x)+x*sin(2*x)=0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \frac {x^{2}}{2}+c_{1} \\ y \left (x \right ) &= -\frac {\sin \left (x \right )}{2}+c_{1} \\ y \left (x \right ) &= -2 \cos \left (x \right )+c_{1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 39

DSolve[2*D[y[x],x]^3-(2*x+4*Sin[x]-Cos[x])*D[y[x],x]^2-(x*Cos[x]-4*x*Sin[x]+Sin[2*x])*D[y[x],x]+x*Sin[2*x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {x^2}{2}+c_1 \\ y(x)\to -2 \cos (x)+c_1 \\ y(x)\to -\frac {\sin (x)}{2}+c_1 \\ \end{align*}