62.12.4 problem Ex 4

Internal problem ID [12850]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 19. Summary. Page 29
Problem number : Ex 4
Date solved : Tuesday, January 28, 2025 at 04:27:16 AM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.276 (sec). Leaf size: 29

dsolve((y(x)-x)^2*diff(y(x),x)=1,y(x), singsol=all)
 

\[ y+\frac {\ln \left (y-x -1\right )}{2}-\frac {\ln \left (y-x +1\right )}{2}-c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.156 (sec). Leaf size: 117

DSolve[(y[x]-x)^2*D[y[x],x]==1,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^x\left (\frac {1}{2 (K[1]-y(x)+1)}-\frac {1}{2 (K[1]-y(x)-1)}\right )dK[1]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {1}{2 (K[1]-K[2]+1)^2}-\frac {1}{2 (K[1]-K[2]-1)^2}\right )dK[1]+\frac {1}{2 (-x+K[2]-1)}-\frac {1}{2 (-x+K[2]+1)}+1\right )dK[2]=c_1,y(x)\right ] \]