83.22.27 problem 30

Internal problem ID [19281]
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 : 30
Date solved : Tuesday, January 28, 2025 at 01:22:23 PM
CAS classification : [_rational]

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

✓ Solution by Maple

Time used: 0.253 (sec). Leaf size: 916

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

\begin{align*} y \left (x \right ) &= -i \left (h -x \right ) \\ y \left (x \right ) &= i \left (h -x \right ) \\ y \left (x \right ) &= -i \left (h +x \right ) \\ y \left (x \right ) &= i \left (h +x \right ) \\ y \left (x \right ) &= 0 \\ -\int _{\textit {\_b}}^{x}\frac {-y \left (x \right )^{2}-h^{2}+\textit {\_a}^{2}-\sqrt {\left (y \left (x \right )^{2}+\left (h +\textit {\_a} \right )^{2}\right ) \left (y \left (x \right )^{2}+\left (h -\textit {\_a} \right )^{2}\right )}}{\left (\left (-y \left (x \right )^{2}-\textit {\_a}^{2}+h^{2}\right ) \sqrt {\left (y \left (x \right )^{2}+\left (h +\textit {\_a} \right )^{2}\right ) \left (y \left (x \right )^{2}+\left (h -\textit {\_a} \right )^{2}\right )}+\left (y \left (x \right )^{2}+\left (h +\textit {\_a} \right )^{2}\right ) \left (y \left (x \right )^{2}+\left (h -\textit {\_a} \right )^{2}\right )\right ) \textit {\_a}}d \textit {\_a} -2 \left (\int _{}^{y \left (x \right )}\frac {4 \left (\frac {1}{4}+\left (\left (-\textit {\_f}^{2}+h^{2}-x^{2}\right ) \sqrt {\left (\textit {\_f}^{2}+\left (h +x \right )^{2}\right ) \left (\textit {\_f}^{2}+\left (h -x \right )^{2}\right )}+\left (\textit {\_f}^{2}+\left (h +x \right )^{2}\right ) \left (\textit {\_f}^{2}+\left (h -x \right )^{2}\right )\right ) \left (\int _{\textit {\_b}}^{x}-\frac {\textit {\_a} \left (\left (-\textit {\_a}^{2}-\textit {\_f}^{2}+h^{2}\right ) \sqrt {\textit {\_a}^{4}+\left (2 \textit {\_f}^{2}-2 h^{2}\right ) \textit {\_a}^{2}+\left (\textit {\_f}^{2}+h^{2}\right )^{2}}+\textit {\_a}^{4}+\left (2 \textit {\_f}^{2}-2 h^{2}\right ) \textit {\_a}^{2}+h^{4}+\textit {\_f}^{4}\right )}{\sqrt {\textit {\_a}^{4}+\left (2 \textit {\_f}^{2}-2 h^{2}\right ) \textit {\_a}^{2}+\left (\textit {\_f}^{2}+h^{2}\right )^{2}}\, {\left (\left (-\textit {\_a}^{2}-\textit {\_f}^{2}+h^{2}\right ) \sqrt {\textit {\_a}^{4}+\left (2 \textit {\_f}^{2}-2 h^{2}\right ) \textit {\_a}^{2}+\left (\textit {\_f}^{2}+h^{2}\right )^{2}}+\textit {\_a}^{4}+\left (2 \textit {\_f}^{2}-2 h^{2}\right ) \textit {\_a}^{2}+\left (\textit {\_f}^{2}+h^{2}\right )^{2}\right )}^{2}}d \textit {\_a} \right )\right ) \textit {\_f}}{\left (-\textit {\_f}^{2}+h^{2}-x^{2}\right ) \sqrt {\left (\textit {\_f}^{2}+\left (h +x \right )^{2}\right ) \left (\textit {\_f}^{2}+\left (h -x \right )^{2}\right )}+\left (\textit {\_f}^{2}+\left (h +x \right )^{2}\right ) \left (\textit {\_f}^{2}+\left (h -x \right )^{2}\right )}d \textit {\_f} \right )+c_{1} &= 0 \\ -\int _{\textit {\_b}}^{x}\frac {-y \left (x \right )^{2}-h^{2}+\textit {\_a}^{2}+\sqrt {\left (y \left (x \right )^{2}+\left (h +\textit {\_a} \right )^{2}\right ) \left (y \left (x \right )^{2}+\left (h -\textit {\_a} \right )^{2}\right )}}{\textit {\_a} \left (\left (y \left (x \right )^{2}+\textit {\_a}^{2}-h^{2}\right ) \sqrt {\left (y \left (x \right )^{2}+\left (h +\textit {\_a} \right )^{2}\right ) \left (y \left (x \right )^{2}+\left (h -\textit {\_a} \right )^{2}\right )}+\left (y \left (x \right )^{2}+\left (h +\textit {\_a} \right )^{2}\right ) \left (y \left (x \right )^{2}+\left (h -\textit {\_a} \right )^{2}\right )\right )}d \textit {\_a} -2 \left (\int _{}^{y \left (x \right )}\frac {4 \textit {\_f} \left (\frac {1}{4}+\left (\left (\textit {\_f}^{2}-h^{2}+x^{2}\right ) \sqrt {\left (\textit {\_f}^{2}+\left (h +x \right )^{2}\right ) \left (\textit {\_f}^{2}+\left (h -x \right )^{2}\right )}+\left (\textit {\_f}^{2}+\left (h +x \right )^{2}\right ) \left (\textit {\_f}^{2}+\left (h -x \right )^{2}\right )\right ) \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a} \left (\left (\textit {\_a}^{2}+\textit {\_f}^{2}-h^{2}\right ) \sqrt {\textit {\_a}^{4}+\left (2 \textit {\_f}^{2}-2 h^{2}\right ) \textit {\_a}^{2}+\left (\textit {\_f}^{2}+h^{2}\right )^{2}}+\textit {\_a}^{4}+\left (2 \textit {\_f}^{2}-2 h^{2}\right ) \textit {\_a}^{2}+h^{4}+\textit {\_f}^{4}\right )}{\sqrt {\textit {\_a}^{4}+\left (2 \textit {\_f}^{2}-2 h^{2}\right ) \textit {\_a}^{2}+\left (\textit {\_f}^{2}+h^{2}\right )^{2}}\, {\left (\left (\textit {\_a}^{2}+\textit {\_f}^{2}-h^{2}\right ) \sqrt {\textit {\_a}^{4}+\left (2 \textit {\_f}^{2}-2 h^{2}\right ) \textit {\_a}^{2}+\left (\textit {\_f}^{2}+h^{2}\right )^{2}}+\textit {\_a}^{4}+\left (2 \textit {\_f}^{2}-2 h^{2}\right ) \textit {\_a}^{2}+\left (\textit {\_f}^{2}+h^{2}\right )^{2}\right )}^{2}}d \textit {\_a} \right )\right )}{\left (\textit {\_f}^{2}-h^{2}+x^{2}\right ) \sqrt {\left (\textit {\_f}^{2}+\left (h +x \right )^{2}\right ) \left (\textit {\_f}^{2}+\left (h -x \right )^{2}\right )}+\left (\textit {\_f}^{2}+\left (h +x \right )^{2}\right ) \left (\textit {\_f}^{2}+\left (h -x \right )^{2}\right )}d \textit {\_f} \right )+c_{1} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.348 (sec). Leaf size: 75

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

\begin{align*} y(x)\to \sqrt {c_1 \left (x^2-\frac {h^2}{1+c_1}\right )} \\ y(x)\to -i (h-x) \\ y(x)\to i (h-x) \\ y(x)\to -i (h+x) \\ y(x)\to i (h+x) \\ \end{align*}