40.14.12 problem 33
Internal
problem
ID
[6783]
Book
:
Schaums
Outline.
Theory
and
problems
of
Differential
Equations,
1st
edition.
Frank
Ayres.
McGraw
Hill
1952
Section
:
Chapter
19.
Linear
equations
with
variable
coefficients
(Misc.
types).
Supplemetary
problems.
Page
132
Problem
number
:
33
Date
solved
:
Tuesday, January 28, 2025 at 03:10:33 PM
CAS
classification
:
[[_3rd_order, _exact, _nonlinear]]
\begin{align*} \left (1+2 y+3 y^{2}\right ) y^{\prime \prime \prime }+6 y^{\prime } \left (y^{\prime \prime }+{y^{\prime }}^{2}+3 y y^{\prime \prime }\right )&=x \end{align*}
✓ Solution by Maple
Time used: 0.030 (sec). Leaf size: 1106
dsolve((1+2*y(x)+3*y(x)^2)*diff(y(x),x$3)+6*diff(y(x),x)*( diff(y(x),x$2)+diff(y(x),x)^2+3*y(x)*diff(y(x),x$2) )=x,y(x), singsol=all)
\begin{align*}
y &= \frac {\left (224+36 x^{4}-432 c_1 \,x^{2}-432 c_1^{2}-864 c_2 x +864 c_3 +12 \sqrt {9 x^{8}-216 c_1 \,x^{6}+1080 c_1^{2} x^{4}-432 c_2 \,x^{5}+2592 c_1^{3} x^{2}+5184 c_1 c_2 \,x^{3}+432 c_3 \,x^{4}+1296 c_1^{4}+5184 c_1^{2} c_2 x -5184 c_1 c_3 \,x^{2}+5184 c_2^{2} x^{2}+112 x^{4}-5184 c_1^{2} c_3 -1344 c_1 \,x^{2}-10368 c_2 c_3 x -1344 c_1^{2}-2688 c_2 x +5184 c_3^{2}+2688 c_3 +576}\right )^{{1}/{3}}}{12}-\frac {8}{3 \left (224+36 x^{4}-432 c_1 \,x^{2}-432 c_1^{2}-864 c_2 x +864 c_3 +12 \sqrt {9 x^{8}-216 c_1 \,x^{6}+1080 c_1^{2} x^{4}-432 c_2 \,x^{5}+2592 c_1^{3} x^{2}+5184 c_1 c_2 \,x^{3}+432 c_3 \,x^{4}+1296 c_1^{4}+5184 c_1^{2} c_2 x -5184 c_1 c_3 \,x^{2}+5184 c_2^{2} x^{2}+112 x^{4}-5184 c_1^{2} c_3 -1344 c_1 \,x^{2}-10368 c_2 c_3 x -1344 c_1^{2}-2688 c_2 x +5184 c_3^{2}+2688 c_3 +576}\right )^{{1}/{3}}}-\frac {1}{3} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (224+36 x^{4}-432 c_1 \,x^{2}-432 c_1^{2}-864 c_2 x +864 c_3 +12 \sqrt {9 x^{8}-216 c_1 \,x^{6}-432 c_2 \,x^{5}+\left (1080 c_1^{2}+432 c_3 +112\right ) x^{4}+5184 c_1 c_2 \,x^{3}+\left (2592 c_1^{3}+\left (-5184 c_3 -1344\right ) c_1 +5184 c_2^{2}\right ) x^{2}+5184 c_2 \left (c_1^{2}-2 c_3 -\frac {14}{27}\right ) x +1296 c_1^{4}+\left (-5184 c_3 -1344\right ) c_1^{2}+5184 c_3^{2}+2688 c_3 +576}\right )^{{2}/{3}}+32 i \sqrt {3}+8 \left (224+36 x^{4}-432 c_1 \,x^{2}-432 c_1^{2}-864 c_2 x +864 c_3 +12 \sqrt {9 x^{8}-216 c_1 \,x^{6}-432 c_2 \,x^{5}+\left (1080 c_1^{2}+432 c_3 +112\right ) x^{4}+5184 c_1 c_2 \,x^{3}+\left (2592 c_1^{3}+\left (-5184 c_3 -1344\right ) c_1 +5184 c_2^{2}\right ) x^{2}+5184 c_2 \left (c_1^{2}-2 c_3 -\frac {14}{27}\right ) x +1296 c_1^{4}+\left (-5184 c_3 -1344\right ) c_1^{2}+5184 c_3^{2}+2688 c_3 +576}\right )^{{1}/{3}}-32}{24 \left (224+36 x^{4}-432 c_1 \,x^{2}-432 c_1^{2}-864 c_2 x +864 c_3 +12 \sqrt {9 x^{8}-216 c_1 \,x^{6}-432 c_2 \,x^{5}+\left (1080 c_1^{2}+432 c_3 +112\right ) x^{4}+5184 c_1 c_2 \,x^{3}+\left (2592 c_1^{3}+\left (-5184 c_3 -1344\right ) c_1 +5184 c_2^{2}\right ) x^{2}+5184 c_2 \left (c_1^{2}-2 c_3 -\frac {14}{27}\right ) x +1296 c_1^{4}+\left (-5184 c_3 -1344\right ) c_1^{2}+5184 c_3^{2}+2688 c_3 +576}\right )^{{1}/{3}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (224+36 x^{4}-432 c_1 \,x^{2}-432 c_1^{2}-864 c_2 x +864 c_3 +12 \sqrt {9 x^{8}-216 c_1 \,x^{6}-432 c_2 \,x^{5}+\left (1080 c_1^{2}+432 c_3 +112\right ) x^{4}+5184 c_1 c_2 \,x^{3}+\left (2592 c_1^{3}+\left (-5184 c_3 -1344\right ) c_1 +5184 c_2^{2}\right ) x^{2}+5184 c_2 \left (c_1^{2}-2 c_3 -\frac {14}{27}\right ) x +1296 c_1^{4}+\left (-5184 c_3 -1344\right ) c_1^{2}+5184 c_3^{2}+2688 c_3 +576}\right )^{{2}/{3}}+32 i \sqrt {3}-8 \left (224+36 x^{4}-432 c_1 \,x^{2}-432 c_1^{2}-864 c_2 x +864 c_3 +12 \sqrt {9 x^{8}-216 c_1 \,x^{6}-432 c_2 \,x^{5}+\left (1080 c_1^{2}+432 c_3 +112\right ) x^{4}+5184 c_1 c_2 \,x^{3}+\left (2592 c_1^{3}+\left (-5184 c_3 -1344\right ) c_1 +5184 c_2^{2}\right ) x^{2}+5184 c_2 \left (c_1^{2}-2 c_3 -\frac {14}{27}\right ) x +1296 c_1^{4}+\left (-5184 c_3 -1344\right ) c_1^{2}+5184 c_3^{2}+2688 c_3 +576}\right )^{{1}/{3}}+32}{24 \left (224+36 x^{4}-432 c_1 \,x^{2}-432 c_1^{2}-864 c_2 x +864 c_3 +12 \sqrt {9 x^{8}-216 c_1 \,x^{6}-432 c_2 \,x^{5}+\left (1080 c_1^{2}+432 c_3 +112\right ) x^{4}+5184 c_1 c_2 \,x^{3}+\left (2592 c_1^{3}+\left (-5184 c_3 -1344\right ) c_1 +5184 c_2^{2}\right ) x^{2}+5184 c_2 \left (c_1^{2}-2 c_3 -\frac {14}{27}\right ) x +1296 c_1^{4}+\left (-5184 c_3 -1344\right ) c_1^{2}+5184 c_3^{2}+2688 c_3 +576}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.349 (sec). Leaf size: 523
DSolve[(1+2*y[x]+3*y[x]^2)*D[y[x],{x,3}]+6*D[y[x],x]*( D[y[x],{x,2}]+D[y[x],x]^2+3*y[x]*D[y[x],{x,2}] )==x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {2^{2/3} \left (9 x^4+108 c_1 x^2+\sqrt {2048+\left (9 x^4+108 c_1 x^2+27 c_3 x+56+216 c_2\right ){}^2}+27 c_3 x+56+216 c_2\right ){}^{2/3}-4 \sqrt [3]{9 x^4+108 c_1 x^2+\sqrt {2048+\left (9 x^4+108 c_1 x^2+27 c_3 x+56+216 c_2\right ){}^2}+27 c_3 x+56+216 c_2}-16 \sqrt [3]{2}}{12 \sqrt [3]{9 x^4+108 c_1 x^2+\sqrt {2048+\left (9 x^4+108 c_1 x^2+27 c_3 x+56+216 c_2\right ){}^2}+27 c_3 x+56+216 c_2}} \\
y(x)\to \frac {1}{24} \left (i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{9 x^4+108 c_1 x^2+\sqrt {2048+\left (9 x^4+108 c_1 x^2+27 c_3 x+56+216 c_2\right ){}^2}+27 c_3 x+56+216 c_2}+\frac {16 \sqrt [3]{2} \left (1+i \sqrt {3}\right )}{\sqrt [3]{9 x^4+108 c_1 x^2+\sqrt {2048+\left (9 x^4+108 c_1 x^2+27 c_3 x+56+216 c_2\right ){}^2}+27 c_3 x+56+216 c_2}}-8\right ) \\
y(x)\to \frac {1}{24} \left (-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{9 x^4+108 c_1 x^2+\sqrt {2048+\left (9 x^4+108 c_1 x^2+27 c_3 x+56+216 c_2\right ){}^2}+27 c_3 x+56+216 c_2}+\frac {16 \sqrt [3]{2} \left (1-i \sqrt {3}\right )}{\sqrt [3]{9 x^4+108 c_1 x^2+\sqrt {2048+\left (9 x^4+108 c_1 x^2+27 c_3 x+56+216 c_2\right ){}^2}+27 c_3 x+56+216 c_2}}-8\right ) \\
\end{align*}