Internal problem ID [5319]
Internal file name [OUTPUT/4810_Friday_February_02_2024_05_13_54_AM_49569275/index.tex]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 6. Equations of first order and first degree (Linear equations). Supplemetary
problems. Page 39
Problem number: 23 (b).
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational]
Unable to solve or complete the solution.
\[ \boxed {4 y y^{\prime } x^{2}-3 x \left (3 y^{2}+2\right )-2 \left (3 y^{2}+2\right )^{3}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 4 y y^{\prime } x^{2}-3 x \left (3 y^{2}+2\right )-2 \left (3 y^{2}+2\right )^{3}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {3 x \left (3 y^{2}+2\right )+2 \left (3 y^{2}+2\right )^{3}}{4 y x^{2}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE differential order: 1; trying a linearization to 2nd order --- trying a change of variables {x -> y(x), y(x) -> x} differential order: 1; trying a linearization to 2nd order trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 2`[0, (2*x*y^4+x*y^6+8/27*x+8/81*x^2+4/3*x*y^2+4/27*x^2*y^2)/x^2/y]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 175
dsolve(4*x^2*y(x)*diff(y(x),x)=3*x*(3*y(x)^2+2)+2*(3*y(x)^2+2)^3,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {\sqrt {6}\, \sqrt {\frac {-3 c_{1} x^{8}+\sqrt {-3 \left (c_{1} x^{8}-\frac {1}{3}\right ) c_{1} x^{9}}+1}{3 c_{1} x^{8}-1}}}{3} \\ y \left (x \right ) &= \frac {\sqrt {6}\, \sqrt {\frac {-3 c_{1} x^{8}+\sqrt {-3 \left (c_{1} x^{8}-\frac {1}{3}\right ) c_{1} x^{9}}+1}{3 c_{1} x^{8}-1}}}{3} \\ y \left (x \right ) &= -\frac {\sqrt {\frac {-18 c_{1} x^{8}-6 \sqrt {-3 \left (c_{1} x^{8}-\frac {1}{3}\right ) c_{1} x^{9}}+6}{3 c_{1} x^{8}-1}}}{3} \\ y \left (x \right ) &= \frac {\sqrt {\frac {-18 c_{1} x^{8}-6 \sqrt {-3 \left (c_{1} x^{8}-\frac {1}{3}\right ) c_{1} x^{9}}+6}{3 c_{1} x^{8}-1}}}{3} \\ \end{align*}
✓ Solution by Mathematica
Time used: 19.518 (sec). Leaf size: 277
DSolve[4*x^2*y[x]*y'[x]==3*x*(3*y[x]^2+2)+2*(3*y[x]^2+2)^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{3} \sqrt {2} \sqrt {-\frac {3 x^8+\sqrt {3} \sqrt {-x^9 \left (x^8+72 c_1\right )}+216 c_1}{x^8+72 c_1}} \\ y(x)\to \frac {1}{3} \sqrt {2} \sqrt {-\frac {3 x^8+\sqrt {3} \sqrt {-x^9 \left (x^8+72 c_1\right )}+216 c_1}{x^8+72 c_1}} \\ y(x)\to -\frac {1}{3} \sqrt {2} \sqrt {\frac {-3 x^8+\sqrt {3} \sqrt {-x^9 \left (x^8+72 c_1\right )}-216 c_1}{x^8+72 c_1}} \\ y(x)\to \frac {1}{3} \sqrt {2} \sqrt {\frac {-3 x^8+\sqrt {3} \sqrt {-x^9 \left (x^8+72 c_1\right )}-216 c_1}{x^8+72 c_1}} \\ y(x)\to -i \sqrt {\frac {2}{3}} \\ y(x)\to i \sqrt {\frac {2}{3}} \\ \end{align*}