82.12.32 problem Ex. 35

Internal problem ID [18797]
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. 35
Date solved : Tuesday, January 28, 2025 at 12:17:50 PM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Solution by Maple

Time used: 0.242 (sec). Leaf size: 43

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

\[ y \left (x \right ) = \frac {\operatorname {RootOf}\left (\textit {\_Z}^{60} c_{1} +15 \textit {\_Z}^{40} c_{1} -64 x^{5} \textit {\_Z}^{35}+75 \textit {\_Z}^{20} c_{1} +125 c_{1} \right )^{20}+5}{4 x^{2}} \]

✓ Solution by Mathematica

Time used: 10.850 (sec). Leaf size: 997

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

\begin{align*} y(x)\to \text {Root}\left [\text {$\#$1}^{12} \left (-e^{20 c_1}\right )-16384 \text {$\#$1}^7 x^{10}+143360 \text {$\#$1}^6 x^8-537600 \text {$\#$1}^5 x^6+1120000 \text {$\#$1}^4 x^4-1400000 \text {$\#$1}^3 x^2+1050000 \text {$\#$1}^2-\frac {437500 \text {$\#$1}}{x^2}+\frac {78125}{x^4}\&,1\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^{12} \left (-e^{20 c_1}\right )-16384 \text {$\#$1}^7 x^{10}+143360 \text {$\#$1}^6 x^8-537600 \text {$\#$1}^5 x^6+1120000 \text {$\#$1}^4 x^4-1400000 \text {$\#$1}^3 x^2+1050000 \text {$\#$1}^2-\frac {437500 \text {$\#$1}}{x^2}+\frac {78125}{x^4}\&,2\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^{12} \left (-e^{20 c_1}\right )-16384 \text {$\#$1}^7 x^{10}+143360 \text {$\#$1}^6 x^8-537600 \text {$\#$1}^5 x^6+1120000 \text {$\#$1}^4 x^4-1400000 \text {$\#$1}^3 x^2+1050000 \text {$\#$1}^2-\frac {437500 \text {$\#$1}}{x^2}+\frac {78125}{x^4}\&,3\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^{12} \left (-e^{20 c_1}\right )-16384 \text {$\#$1}^7 x^{10}+143360 \text {$\#$1}^6 x^8-537600 \text {$\#$1}^5 x^6+1120000 \text {$\#$1}^4 x^4-1400000 \text {$\#$1}^3 x^2+1050000 \text {$\#$1}^2-\frac {437500 \text {$\#$1}}{x^2}+\frac {78125}{x^4}\&,4\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^{12} \left (-e^{20 c_1}\right )-16384 \text {$\#$1}^7 x^{10}+143360 \text {$\#$1}^6 x^8-537600 \text {$\#$1}^5 x^6+1120000 \text {$\#$1}^4 x^4-1400000 \text {$\#$1}^3 x^2+1050000 \text {$\#$1}^2-\frac {437500 \text {$\#$1}}{x^2}+\frac {78125}{x^4}\&,5\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^{12} \left (-e^{20 c_1}\right )-16384 \text {$\#$1}^7 x^{10}+143360 \text {$\#$1}^6 x^8-537600 \text {$\#$1}^5 x^6+1120000 \text {$\#$1}^4 x^4-1400000 \text {$\#$1}^3 x^2+1050000 \text {$\#$1}^2-\frac {437500 \text {$\#$1}}{x^2}+\frac {78125}{x^4}\&,6\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^{12} \left (-e^{20 c_1}\right )-16384 \text {$\#$1}^7 x^{10}+143360 \text {$\#$1}^6 x^8-537600 \text {$\#$1}^5 x^6+1120000 \text {$\#$1}^4 x^4-1400000 \text {$\#$1}^3 x^2+1050000 \text {$\#$1}^2-\frac {437500 \text {$\#$1}}{x^2}+\frac {78125}{x^4}\&,7\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^{12} \left (-e^{20 c_1}\right )-16384 \text {$\#$1}^7 x^{10}+143360 \text {$\#$1}^6 x^8-537600 \text {$\#$1}^5 x^6+1120000 \text {$\#$1}^4 x^4-1400000 \text {$\#$1}^3 x^2+1050000 \text {$\#$1}^2-\frac {437500 \text {$\#$1}}{x^2}+\frac {78125}{x^4}\&,8\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^{12} \left (-e^{20 c_1}\right )-16384 \text {$\#$1}^7 x^{10}+143360 \text {$\#$1}^6 x^8-537600 \text {$\#$1}^5 x^6+1120000 \text {$\#$1}^4 x^4-1400000 \text {$\#$1}^3 x^2+1050000 \text {$\#$1}^2-\frac {437500 \text {$\#$1}}{x^2}+\frac {78125}{x^4}\&,9\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^{12} \left (-e^{20 c_1}\right )-16384 \text {$\#$1}^7 x^{10}+143360 \text {$\#$1}^6 x^8-537600 \text {$\#$1}^5 x^6+1120000 \text {$\#$1}^4 x^4-1400000 \text {$\#$1}^3 x^2+1050000 \text {$\#$1}^2-\frac {437500 \text {$\#$1}}{x^2}+\frac {78125}{x^4}\&,10\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^{12} \left (-e^{20 c_1}\right )-16384 \text {$\#$1}^7 x^{10}+143360 \text {$\#$1}^6 x^8-537600 \text {$\#$1}^5 x^6+1120000 \text {$\#$1}^4 x^4-1400000 \text {$\#$1}^3 x^2+1050000 \text {$\#$1}^2-\frac {437500 \text {$\#$1}}{x^2}+\frac {78125}{x^4}\&,11\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^{12} \left (-e^{20 c_1}\right )-16384 \text {$\#$1}^7 x^{10}+143360 \text {$\#$1}^6 x^8-537600 \text {$\#$1}^5 x^6+1120000 \text {$\#$1}^4 x^4-1400000 \text {$\#$1}^3 x^2+1050000 \text {$\#$1}^2-\frac {437500 \text {$\#$1}}{x^2}+\frac {78125}{x^4}\&,12\right ] \\ \end{align*}