Internal problem ID [5242]
Internal file name [OUTPUT/4733_Friday_February_02_2024_05_10_46_AM_84485799/index.tex]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 4. Equations of first order and first degree (Variable separable). Supplemetary
problems. Page 22
Problem number: 30.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "homogeneousTypeD2"
Maple gives the following as the ode type
[[_homogeneous, `class D`], _rational]
\[ \boxed {y^{2} \left (x^{2}+2\right )+\left (x^{3}+y^{3}\right ) \left (-y^{\prime } x +y\right )=0} \]
Using the change of variables \(y = u \left (x \right ) x\) on the above ode results in new ode in \(u \left (x \right )\) \begin {align*} u \left (x \right )^{2} x^{2} \left (x^{2}+2\right )+\left (x^{3}+u \left (x \right )^{3} x^{3}\right ) \left (-\left (u^{\prime }\left (x \right ) x +u \left (x \right )\right ) x +u \left (x \right ) x \right ) = 0 \end {align*}
In canonical form the ODE is \begin {align*} u' &= F(x,u)\\ &= f( x) g(u)\\ &= \frac {u^{2} \left (x^{2}+2\right )}{\left (u^{3}+1\right ) x^{3}} \end {align*}
Where \(f(x)=\frac {x^{2}+2}{x^{3}}\) and \(g(u)=\frac {u^{2}}{u^{3}+1}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {u^{2}}{u^{3}+1}} \,du &= \frac {x^{2}+2}{x^{3}} \,d x \\ \int { \frac {1}{\frac {u^{2}}{u^{3}+1}} \,du} &= \int {\frac {x^{2}+2}{x^{3}} \,d x} \\ \frac {u^{3}-2}{2 u}&=-\frac {1}{x^{2}}+\ln \left (x \right )+c_{2} \\ \end{align*} The solution is \[ \frac {u \left (x \right )^{3}-2}{2 u \left (x \right )}+\frac {1}{x^{2}}-\ln \left (x \right )-c_{2} = 0 \] Replacing \(u(x)\) in the above solution by \(\frac {y}{x}\) results in the solution for \(y\) in implicit form \begin {align*} \frac {\left (\frac {y^{3}}{x^{3}}-2\right ) x}{2 y}+\frac {1}{x^{2}}-\ln \left (x \right )-c_{2} = 0\\ \frac {\left (\frac {y^{3}}{x^{3}}-2\right ) x}{2 y}+\frac {1}{x^{2}}-\ln \left (x \right )-c_{2} = 0 \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \frac {\left (\frac {y^{3}}{x^{3}}-2\right ) x}{2 y}+\frac {1}{x^{2}}-\ln \left (x \right )-c_{2} &= 0 \\ \end{align*}
Verification of solutions
\[ \frac {\left (\frac {y^{3}}{x^{3}}-2\right ) x}{2 y}+\frac {1}{x^{2}}-\ln \left (x \right )-c_{2} = 0 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{2} \left (x^{2}+2\right )+\left (x^{3}+y^{3}\right ) \left (-y^{\prime } x +y\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y \left (y^{3}+y x^{2}+x^{3}+2 y\right )}{x \left (x^{3}+y^{3}\right )} \end {array} \]
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 homogeneous D <- homogeneous successful`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 659
dsolve(y(x)^2*(x^2+2)+(x^3+y(x)^3)*(y(x)-x*diff(y(x),x))=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {6 \ln \left (x \right ) x^{2}+6 c_{1} x^{2}+\left (27 x^{3}+3 \sqrt {-24 c_{1}^{3} x^{6}-72 c_{1}^{2} x^{6} \ln \left (x \right )+72 c_{1}^{2} x^{4}-72 c_{1} x^{6} \ln \left (x \right )^{2}+144 c_{1} x^{4} \ln \left (x \right )-72 c_{1} x^{2}-24 \ln \left (x \right )^{3} x^{6}+72 \ln \left (x \right )^{2} x^{4}-72 \ln \left (x \right ) x^{2}+24+81 x^{6}}\right )^{\frac {2}{3}}-6}{3 \left (27 x^{3}+3 \sqrt {-24 c_{1}^{3} x^{6}-72 c_{1}^{2} x^{6} \ln \left (x \right )+72 c_{1}^{2} x^{4}-72 c_{1} x^{6} \ln \left (x \right )^{2}+144 c_{1} x^{4} \ln \left (x \right )-72 c_{1} x^{2}-24 \ln \left (x \right )^{3} x^{6}+72 \ln \left (x \right )^{2} x^{4}-72 \ln \left (x \right ) x^{2}+24+81 x^{6}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\frac {\left (-1-i \sqrt {3}\right ) \left (27 x^{3}+3 \sqrt {-24 c_{1}^{3} x^{6}-72 c_{1}^{2} x^{6} \ln \left (x \right )+72 c_{1}^{2} x^{4}-72 c_{1} x^{6} \ln \left (x \right )^{2}+144 c_{1} x^{4} \ln \left (x \right )-72 c_{1} x^{2}-24 \ln \left (x \right )^{3} x^{6}+72 \ln \left (x \right )^{2} x^{4}-72 \ln \left (x \right ) x^{2}+24+81 x^{6}}\right )^{\frac {2}{3}}}{6}+\left (i \sqrt {3}-1\right ) \left (c_{1} x^{2}+\ln \left (x \right ) x^{2}-1\right )}{\left (27 x^{3}+3 \sqrt {-24 c_{1}^{3} x^{6}-72 c_{1}^{2} x^{6} \ln \left (x \right )+72 c_{1}^{2} x^{4}-72 c_{1} x^{6} \ln \left (x \right )^{2}+144 c_{1} x^{4} \ln \left (x \right )-72 c_{1} x^{2}-24 \ln \left (x \right )^{3} x^{6}+72 \ln \left (x \right )^{2} x^{4}-72 \ln \left (x \right ) x^{2}+24+81 x^{6}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (27 x^{3}+3 \sqrt {-24 c_{1}^{3} x^{6}-72 c_{1}^{2} x^{6} \ln \left (x \right )+72 c_{1}^{2} x^{4}-72 c_{1} x^{6} \ln \left (x \right )^{2}+144 c_{1} x^{4} \ln \left (x \right )-72 c_{1} x^{2}-24 \ln \left (x \right )^{3} x^{6}+72 \ln \left (x \right )^{2} x^{4}-72 \ln \left (x \right ) x^{2}+24+81 x^{6}}\right )^{\frac {2}{3}}}{6}+\left (-1-i \sqrt {3}\right ) \left (c_{1} x^{2}+\ln \left (x \right ) x^{2}-1\right )}{\left (27 x^{3}+3 \sqrt {-24 c_{1}^{3} x^{6}-72 c_{1}^{2} x^{6} \ln \left (x \right )+72 c_{1}^{2} x^{4}-72 c_{1} x^{6} \ln \left (x \right )^{2}+144 c_{1} x^{4} \ln \left (x \right )-72 c_{1} x^{2}-24 \ln \left (x \right )^{3} x^{6}+72 \ln \left (x \right )^{2} x^{4}-72 \ln \left (x \right ) x^{2}+24+81 x^{6}}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 54.35 (sec). Leaf size: 396
DSolve[y[x]^2*(x^2+2)+(x^3+y[x]^3)*(y[x]-x*y'[x])==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {6 x^2 \log (x)+6 c_1 x^2+3^{2/3} \left (9 x^3+\frac {1}{3} \sqrt {729 x^6+\left (-6 x^2 \log (x)-6 c_1 x^2+6\right ){}^3}\right ){}^{2/3}-6}{3 \sqrt [3]{3} \sqrt [3]{9 x^3+\frac {1}{3} \sqrt {729 x^6+\left (-6 x^2 \log (x)-6 c_1 x^2+6\right ){}^3}}} \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{9 x^3+\frac {1}{3} \sqrt {729 x^6+\left (-6 x^2 \log (x)-6 c_1 x^2+6\right ){}^3}}}{2\ 3^{2/3}}-\frac {i \sqrt [3]{2} \left (\sqrt {3}-i\right ) \left (x^2 \log (x)+c_1 x^2-1\right )}{\sqrt [3]{54 x^3+2 \sqrt {729 x^6+\left (-6 x^2 \log (x)-6 c_1 x^2+6\right ){}^3}}} \\ y(x)\to \frac {i \sqrt [3]{2} \left (\sqrt {3}+i\right ) \left (x^2 \log (x)+c_1 x^2-1\right )}{\sqrt [3]{54 x^3+2 \sqrt {729 x^6+\left (-6 x^2 \log (x)-6 c_1 x^2+6\right ){}^3}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{54 x^3+2 \sqrt {729 x^6+\left (-6 x^2 \log (x)-6 c_1 x^2+6\right ){}^3}}}{6 \sqrt [3]{2}} \\ \end{align*}