62.12.12 problem Ex 13

Internal problem ID [12858]
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 13
Date solved : Tuesday, January 28, 2025 at 04:28:42 AM
CAS classification : [_linear]

\begin{align*} y^{\prime }+\frac {y}{\left (-x^{2}+1\right )^{{3}/{2}}}&=\frac {x +\sqrt {-x^{2}+1}}{\left (-x^{2}+1\right )^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 56

dsolve(diff(y(x),x)+y(x)/(1-x^2)^(3/2)= (x+(1-x^2)^(1/2))/(1-x^2)^2,y(x), singsol=all)
 

\[ y = \left (\int \frac {{\mathrm e}^{\frac {x}{\sqrt {-x^{2}+1}}} \left (x +\sqrt {-x^{2}+1}\right )}{\left (x +1\right )^{2} \left (x -1\right )^{2}}d x +c_{1} \right ) {\mathrm e}^{-\frac {x}{\sqrt {-x^{2}+1}}} \]

✓ Solution by Mathematica

Time used: 0.132 (sec). Leaf size: 38

DSolve[D[y[x],x]+y[x]/(1-x^2)^(3/2)== (x+(1-x^2)^(1/2))/(1-x^2)^2,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {x}{\sqrt {1-x^2}}+c_1 e^{-\frac {x}{\sqrt {1-x^2}}} \]