83.23.29 problem 29

Internal problem ID [19310]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter V. Singular solutions. Exercise V at page 76
Problem number : 29
Date solved : Tuesday, January 28, 2025 at 01:30:47 PM
CAS classification : [_separable]

\begin{align*} 4 {y^{\prime }}^{2} x^{2} \left (x -1\right )-4 y^{\prime } x y \left (4 x -3\right )+\left (16 x -9\right ) y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.044 (sec). Leaf size: 57

dsolve(4*diff(y(x),x)^2*x^2*(x-1)-4*diff(y(x),x)*x*y(x)*(4*x-3)+(16*x-9)*y(x)^2=0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {c_{1} \sqrt {1-x}\, \sqrt {x -1}\, x^{{3}/{2}}}{\sqrt {x}+1} \\ y \left (x \right ) &= \frac {c_{1} \left (\sqrt {x}+1\right ) \sqrt {x -1}\, x^{{3}/{2}}}{\sqrt {1-x}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.141 (sec). Leaf size: 44

DSolve[4*D[y[x],x]^2*x^2*(x-1)-4*D[y[x],x]*x*y[x]*(4*x-3)+(16*x-9)*y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to c_1 \left (\sqrt {x}+1\right ) x^{3/2} \\ y(x)\to c_1 \left (\sqrt {x}-1\right ) x^{3/2} \\ y(x)\to 0 \\ \end{align*}