75.12.38 problem 312

Internal problem ID [16904]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 12. Miscellaneous problems. Exercises page 93
Problem number : 312
Date solved : Tuesday, January 28, 2025 at 09:40:27 AM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} 4 x^{2} {y^{\prime }}^{2}-y^{2}&=x y^{3} \end{align*}

✓ Solution by Maple

Time used: 0.339 (sec). Leaf size: 1759

dsolve(4*x^2*diff(y(x),x)^2-y(x)^2=x*y(x)^3,y(x), singsol=all)
 

\begin{align*} y &= 0 \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align*}

✓ Solution by Mathematica

Time used: 105.673 (sec). Leaf size: 1401

DSolve[4*x^2*D[y[x],x]^2-y[x]^2==x*y[x]^3,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {Root}\left [\text {$\#$1}^8 \left (x^9-2 e^{3 c_1} x^6+e^{6 c_1} x^3\right )+\text {$\#$1}^7 \left (-24 x^8-120 e^{3 c_1} x^5\right )+\text {$\#$1}^6 \left (252 x^7-444 e^{3 c_1} x^4\right )+\text {$\#$1}^5 \left (-1512 x^6+56 e^{3 c_1} x^3\right )+\text {$\#$1}^4 \left (5670 x^5-66 e^{3 c_1} x^2\right )+\text {$\#$1}^3 \left (-13608 x^4+48 e^{3 c_1} x\right )+\text {$\#$1}^2 \left (20412 x^3-16 e^{3 c_1}\right )-17496 \text {$\#$1} x^2+6561 x\&,1\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^8 \left (x^9-2 e^{3 c_1} x^6+e^{6 c_1} x^3\right )+\text {$\#$1}^7 \left (-24 x^8-120 e^{3 c_1} x^5\right )+\text {$\#$1}^6 \left (252 x^7-444 e^{3 c_1} x^4\right )+\text {$\#$1}^5 \left (-1512 x^6+56 e^{3 c_1} x^3\right )+\text {$\#$1}^4 \left (5670 x^5-66 e^{3 c_1} x^2\right )+\text {$\#$1}^3 \left (-13608 x^4+48 e^{3 c_1} x\right )+\text {$\#$1}^2 \left (20412 x^3-16 e^{3 c_1}\right )-17496 \text {$\#$1} x^2+6561 x\&,2\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^8 \left (x^9-2 e^{3 c_1} x^6+e^{6 c_1} x^3\right )+\text {$\#$1}^7 \left (-24 x^8-120 e^{3 c_1} x^5\right )+\text {$\#$1}^6 \left (252 x^7-444 e^{3 c_1} x^4\right )+\text {$\#$1}^5 \left (-1512 x^6+56 e^{3 c_1} x^3\right )+\text {$\#$1}^4 \left (5670 x^5-66 e^{3 c_1} x^2\right )+\text {$\#$1}^3 \left (-13608 x^4+48 e^{3 c_1} x\right )+\text {$\#$1}^2 \left (20412 x^3-16 e^{3 c_1}\right )-17496 \text {$\#$1} x^2+6561 x\&,3\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^8 \left (x^9-2 e^{3 c_1} x^6+e^{6 c_1} x^3\right )+\text {$\#$1}^7 \left (-24 x^8-120 e^{3 c_1} x^5\right )+\text {$\#$1}^6 \left (252 x^7-444 e^{3 c_1} x^4\right )+\text {$\#$1}^5 \left (-1512 x^6+56 e^{3 c_1} x^3\right )+\text {$\#$1}^4 \left (5670 x^5-66 e^{3 c_1} x^2\right )+\text {$\#$1}^3 \left (-13608 x^4+48 e^{3 c_1} x\right )+\text {$\#$1}^2 \left (20412 x^3-16 e^{3 c_1}\right )-17496 \text {$\#$1} x^2+6561 x\&,4\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^8 \left (x^9-2 e^{3 c_1} x^6+e^{6 c_1} x^3\right )+\text {$\#$1}^7 \left (-24 x^8-120 e^{3 c_1} x^5\right )+\text {$\#$1}^6 \left (252 x^7-444 e^{3 c_1} x^4\right )+\text {$\#$1}^5 \left (-1512 x^6+56 e^{3 c_1} x^3\right )+\text {$\#$1}^4 \left (5670 x^5-66 e^{3 c_1} x^2\right )+\text {$\#$1}^3 \left (-13608 x^4+48 e^{3 c_1} x\right )+\text {$\#$1}^2 \left (20412 x^3-16 e^{3 c_1}\right )-17496 \text {$\#$1} x^2+6561 x\&,5\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^8 \left (x^9-2 e^{3 c_1} x^6+e^{6 c_1} x^3\right )+\text {$\#$1}^7 \left (-24 x^8-120 e^{3 c_1} x^5\right )+\text {$\#$1}^6 \left (252 x^7-444 e^{3 c_1} x^4\right )+\text {$\#$1}^5 \left (-1512 x^6+56 e^{3 c_1} x^3\right )+\text {$\#$1}^4 \left (5670 x^5-66 e^{3 c_1} x^2\right )+\text {$\#$1}^3 \left (-13608 x^4+48 e^{3 c_1} x\right )+\text {$\#$1}^2 \left (20412 x^3-16 e^{3 c_1}\right )-17496 \text {$\#$1} x^2+6561 x\&,6\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^8 \left (x^9-2 e^{3 c_1} x^6+e^{6 c_1} x^3\right )+\text {$\#$1}^7 \left (-24 x^8-120 e^{3 c_1} x^5\right )+\text {$\#$1}^6 \left (252 x^7-444 e^{3 c_1} x^4\right )+\text {$\#$1}^5 \left (-1512 x^6+56 e^{3 c_1} x^3\right )+\text {$\#$1}^4 \left (5670 x^5-66 e^{3 c_1} x^2\right )+\text {$\#$1}^3 \left (-13608 x^4+48 e^{3 c_1} x\right )+\text {$\#$1}^2 \left (20412 x^3-16 e^{3 c_1}\right )-17496 \text {$\#$1} x^2+6561 x\&,7\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^8 \left (x^9-2 e^{3 c_1} x^6+e^{6 c_1} x^3\right )+\text {$\#$1}^7 \left (-24 x^8-120 e^{3 c_1} x^5\right )+\text {$\#$1}^6 \left (252 x^7-444 e^{3 c_1} x^4\right )+\text {$\#$1}^5 \left (-1512 x^6+56 e^{3 c_1} x^3\right )+\text {$\#$1}^4 \left (5670 x^5-66 e^{3 c_1} x^2\right )+\text {$\#$1}^3 \left (-13608 x^4+48 e^{3 c_1} x\right )+\text {$\#$1}^2 \left (20412 x^3-16 e^{3 c_1}\right )-17496 \text {$\#$1} x^2+6561 x\&,8\right ] \\ \end{align*}