82.12.9 problem Ex. 9

Internal problem ID [18774]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter II. Equations of the first order and of the first degree. Examples on chapter II at page 29
Problem number : Ex. 9
Date solved : Tuesday, January 28, 2025 at 12:16:44 PM
CAS classification : [_linear]

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 39

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

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

✓ Solution by Mathematica

Time used: 5.553 (sec). Leaf size: 164

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

\[ y(x)\to \left (\frac {1}{20}+\frac {i}{40}\right ) \left (\frac {5 i \left (1+e^{2 i \arcsin (x)}\right )^2 \operatorname {Hypergeometric2F1}\left (1-\frac {i}{2},2,2-\frac {i}{2},-e^{2 i \arcsin (x)}\right )}{x^2-1}+(1+2 i) \operatorname {Hypergeometric2F1}\left (-\frac {i}{2},1,1-\frac {i}{2},-e^{2 i \arcsin (x)}\right )-e^{2 i \arcsin (x)} \operatorname {Hypergeometric2F1}\left (1,1-\frac {i}{2},2-\frac {i}{2},-e^{2 i \arcsin (x)}\right )+(16-8 i) c_1 e^{-\arcsin (x)}+\frac {(2-i) \left (x^3+4 \sqrt {1-x^2}-x\right )}{\left (1-x^2\right )^{3/2}}\right ) \]