15.20.14 problem 14

Internal problem ID [3322]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 38, page 173
Problem number : 14
Date solved : Monday, January 27, 2025 at 07:33:39 AM
CAS classification : [_dAlembert]

\begin{align*} 2 {y^{\prime }}^{5}+2 x y^{\prime }&=y \end{align*}

✓ Solution by Maple

Time used: 0.040 (sec). Leaf size: 1295

dsolve(2*diff(y(x),x)^5+2*diff(y(x),x)*x=y(x),y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \frac {20 \sqrt {5}\, \sqrt {-\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{1}/{3}} \left (i \sqrt {3}\, \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{2}/{3}}+20 i \sqrt {3}\, x +\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{2}/{3}}-20 x \right )}\, \left (-\frac {3 \left (c_{1} +\frac {\sqrt {20 x^{3}+225 c_{1}^{2}}}{15}\right ) \left (i \sqrt {3}-1\right ) \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{2}/{3}}}{4}+x \left (\left (1+i \sqrt {3}\right ) x \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{1}/{3}}-90 c_{1} -6 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )\right )}{\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{1}/{3}} \left (15000 c_{1} +1000 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )} \\ y \left (x \right ) &= -\frac {20 \sqrt {5}\, \sqrt {-\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{1}/{3}} \left (i \sqrt {3}\, \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{2}/{3}}+20 i \sqrt {3}\, x +\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{2}/{3}}-20 x \right )}\, \left (-\frac {3 \left (c_{1} +\frac {\sqrt {20 x^{3}+225 c_{1}^{2}}}{15}\right ) \left (i \sqrt {3}-1\right ) \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{2}/{3}}}{4}+x \left (\left (1+i \sqrt {3}\right ) x \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{1}/{3}}-90 c_{1} -6 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )\right )}{\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{1}/{3}} \left (15000 c_{1} +1000 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )} \\ y \left (x \right ) &= -\frac {20 \sqrt {5}\, \sqrt {\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{1}/{3}} \left (i \sqrt {3}\, \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{2}/{3}}+20 i \sqrt {3}\, x -\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{2}/{3}}+20 x \right )}\, \left (-\frac {3 \left (1+i \sqrt {3}\right ) \left (c_{1} +\frac {\sqrt {20 x^{3}+225 c_{1}^{2}}}{15}\right ) \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{2}/{3}}}{4}+x \left (x \left (i \sqrt {3}-1\right ) \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{1}/{3}}+90 c_{1} +6 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )\right )}{\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{1}/{3}} \left (15000 c_{1} +1000 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )} \\ y \left (x \right ) &= \frac {20 \sqrt {5}\, \sqrt {\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{1}/{3}} \left (i \sqrt {3}\, \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{2}/{3}}+20 i \sqrt {3}\, x -\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{2}/{3}}+20 x \right )}\, \left (-\frac {3 \left (1+i \sqrt {3}\right ) \left (c_{1} +\frac {\sqrt {20 x^{3}+225 c_{1}^{2}}}{15}\right ) \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{2}/{3}}}{4}+x \left (x \left (i \sqrt {3}-1\right ) \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{1}/{3}}+90 c_{1} +6 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )\right )}{\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{1}/{3}} \left (15000 c_{1} +1000 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )} \\ y \left (x \right ) &= -\frac {\sqrt {10}\, \left (\frac {\left (3 c_{1} +\frac {\sqrt {20 x^{3}+225 c_{1}^{2}}}{5}\right ) \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{2}/{3}}}{4}+x \left (x \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{1}/{3}}+45 c_{1} +3 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )\right ) \sqrt {\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{1}/{3}} \left (\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{2}/{3}}-20 x \right )}}{25 \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{1}/{3}} \left (15 c_{1} +\sqrt {20 x^{3}+225 c_{1}^{2}}\right )} \\ y \left (x \right ) &= \frac {\sqrt {10}\, \left (\frac {\left (3 c_{1} +\frac {\sqrt {20 x^{3}+225 c_{1}^{2}}}{5}\right ) \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{2}/{3}}}{4}+x \left (x \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{1}/{3}}+45 c_{1} +3 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )\right ) \sqrt {\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{1}/{3}} \left (\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{2}/{3}}-20 x \right )}}{25 \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{{1}/{3}} \left (15 c_{1} +\sqrt {20 x^{3}+225 c_{1}^{2}}\right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 2.023 (sec). Leaf size: 2226

DSolve[2*D[y[x],x]^5+2*D[y[x],x]*x==y[x],y[x],x,IncludeSingularSolutions -> True]
 

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