82.14.2 problem Ex. 2

Internal problem ID [18808]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter III. Equations of the first order but not of the first degree. Problems at page 33
Problem number : Ex. 2
Date solved : Tuesday, January 28, 2025 at 12:20:12 PM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 69

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

\begin{align*} y \left (x \right ) &= c \\ x -\int _{}^{y \left (x \right )}\csc \left (\operatorname {RootOf}\left (-\sin \left (\textit {\_Z} \right )^{2} \cos \left (\textit {\_Z} \right )^{2} a^{2}-2 \sin \left (\textit {\_Z} \right ) \cos \left (\textit {\_Z} \right )^{2} \textit {\_a} a -\cos \left (\textit {\_Z} \right )^{2} \textit {\_a}^{2}+2 a c \textit {\_Z} +c^{2}+a^{2} \textit {\_Z}^{2}\right )\right )d \textit {\_a} -c_{1} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.865 (sec). Leaf size: 74

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

\[ \text {Solve}\left [\left \{x=\frac {K[1] (a \arcsin (K[1])+c)}{\sqrt {1-K[1]^2}}+c_1,y(x)=\frac {a \arcsin (K[1])-a K[1] \sqrt {1-K[1]^2}+c}{\sqrt {1-K[1]^2}}\right \},\{y(x),K[1]\}\right ] \]