85.34.12 problem 12
Internal
problem
ID
[22720]
Book
:
Applied
Differential
Equations.
By
Murray
R.
Spiegel.
3rd
edition.
1980.
Pearson.
ISBN
978-0130400970
Section
:
Chapter
two.
First
order
and
simple
higher
order
ordinary
differential
equations.
B
Exercises
at
page
67
Problem
number
:
12
Date
solved
:
Thursday, October 02, 2025 at 09:13:04 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _Chini]
\begin{align*} y^{\prime }&=\sqrt {y}+x \end{align*}
✓ Maple. Time used: 0.007 (sec). Leaf size: 65
ode:=diff(y(x),x) = x+y(x)^(1/2);
dsolve(ode,y(x), singsol=all);
\[
-\frac {2 \,\operatorname {arctanh}\left (2 \sqrt {\frac {y}{x^{2}}}\right )}{3}+\frac {4 \,\operatorname {arctanh}\left (\sqrt {\frac {y}{x^{2}}}\right )}{3}-\frac {2 \ln \left (-\frac {2 \left (x^{2}-y\right )}{x^{2}}\right )}{3}-\frac {\ln \left (-\frac {x^{2}-4 y}{x^{2}}\right )}{3}-2 \ln \left (x \right )+c_1 = 0
\]
✓ Mathematica. Time used: 31.747 (sec). Leaf size: 716
ode=D[y[x],x]==x+Sqrt[y[x]];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {1}{4} \left (3 x^2+\frac {e^{3 c_1} x \left (8+e^{3 c_1} x^3\right )}{\sqrt [3]{-e^{18 c_1} x^6+20 e^{15 c_1} x^3+8 \sqrt {-e^{24 c_1} \left (-1+e^{3 c_1} x^3\right ){}^3}+8 e^{12 c_1}}}+e^{-6 c_1} \sqrt [3]{-e^{18 c_1} x^6+20 e^{15 c_1} x^3+8 \sqrt {-e^{24 c_1} \left (-1+e^{3 c_1} x^3\right ){}^3}+8 e^{12 c_1}}\right )\\ y(x)&\to \frac {1}{72} \left (54 x^2-\frac {9 i \left (\sqrt {3}-i\right ) e^{3 c_1} x \left (8+e^{3 c_1} x^3\right )}{\sqrt [3]{-e^{18 c_1} x^6+20 e^{15 c_1} x^3+8 \sqrt {-e^{24 c_1} \left (-1+e^{3 c_1} x^3\right ){}^3}+8 e^{12 c_1}}}+9 i \left (\sqrt {3}+i\right ) e^{-6 c_1} \sqrt [3]{-e^{18 c_1} x^6+20 e^{15 c_1} x^3+8 \sqrt {-e^{24 c_1} \left (-1+e^{3 c_1} x^3\right ){}^3}+8 e^{12 c_1}}\right )\\ y(x)&\to \frac {1}{72} \left (54 x^2+\frac {9 i \left (\sqrt {3}+i\right ) e^{3 c_1} x \left (8+e^{3 c_1} x^3\right )}{\sqrt [3]{-e^{18 c_1} x^6+20 e^{15 c_1} x^3+8 \sqrt {-e^{24 c_1} \left (-1+e^{3 c_1} x^3\right ){}^3}+8 e^{12 c_1}}}-9 \left (1+i \sqrt {3}\right ) e^{-6 c_1} \sqrt [3]{-e^{18 c_1} x^6+20 e^{15 c_1} x^3+8 \sqrt {-e^{24 c_1} \left (-1+e^{3 c_1} x^3\right ){}^3}+8 e^{12 c_1}}\right )\\ y(x)&\to \frac {-\left (-x^6\right )^{2/3}+3 x^4+\sqrt [3]{-x^6} x^2}{4 x^2}\\ y(x)&\to \frac {\left (1+i \sqrt {3}\right ) \left (-x^6\right )^{2/3}+6 x^4+i \left (\sqrt {3}+i\right ) \sqrt [3]{-x^6} x^2}{8 x^2}\\ y(x)&\to \frac {1}{8} x^2 \left (\frac {\left (1+i \sqrt {3}\right ) x^4}{\left (-x^6\right )^{2/3}}+\frac {i \left (\sqrt {3}+i\right ) x^2}{\sqrt [3]{-x^6}}+6\right ) \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x - sqrt(y(x)) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -x - sqrt(y(x)) + Derivative(y(x), x) cannot be solved by the li