Internal problem ID [8481]
Internal file name [OUTPUT/7414_Sunday_June_05_2022_10_54_43_PM_50566299/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 145.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "abelFirstKind"
Maple gives the following as the ode type
[_rational, _Abel]
Unable to solve or complete the solution.
\[ \boxed {x^{2} y^{\prime }+a y^{3}-y^{2} a \,x^{2}=0} \]
This is Abel first kind ODE, it has the form \[ y^{\prime }= f_0(x)+f_1(x) y +f_2(x)y^{2}+f_3(x)y^{3} \] Comparing the above to given ODE which is \begin {align*} y^{\prime }&=-\frac {a y^{3}}{x^{2}}+a y^{2}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= 0\\ f_1(x) &= 0\\ f_2(x) &= a\\ f_3(x) &= -\frac {a}{x^{2}} \end {align*}
Since \(f_2(x)=a\) is not zero, then the first step is to apply the following transformation to remove \(f_2\). Let \(y = u(x) - \frac {f_2}{3 f_3}\) or \begin {align*} y &= u(x) - \left ( \frac {a}{-\frac {3 a}{x^{2}}} \right ) \\ &= u \left (x \right )+\frac {x^{2}}{3} \end {align*}
The above transformation applied to (1) gives a new ODE as \begin {align*} u^{\prime }\left (x \right ) = \frac {2 a \,x^{4}}{27}+\frac {x^{2} a u \left (x \right )}{3}-\frac {a u \left (x \right )^{3}}{x^{2}}-\frac {2 x}{3}\tag {2} \end {align*}
This is Abel first kind ODE, it has the form \[ u^{\prime }\left (x \right )= f_0(x)+f_1(x) u \left (x \right ) +f_2(x)u \left (x \right )^{2}+f_3(x)u \left (x \right )^{3} \] Comparing the above to given ODE which is \begin {align*} u^{\prime }\left (x \right )&=-\frac {a u \left (x \right )^{3}}{x^{2}}+\frac {x^{2} a u \left (x \right )}{3}-\frac {-2 a \,x^{6}+18 x^{3}}{27 x^{2}}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= \frac {2}{27} a \,x^{4}-\frac {2}{3} x\\ f_1(x) &= \frac {a \,x^{2}}{3}\\ f_2(x) &= 0\\ f_3(x) &= -\frac {a}{x^{2}} \end {align*}
Since \(f_2(x)=0\) then we check the Abel invariant to see if it depends on \(x\) or not. The Abel invariant is given by \begin {align*} -\frac {f_{1}^{3}}{f_{0}^{2} f_{3}} \end {align*}
Which when evaluating gives \begin {align*} -\frac {{\left (\frac {\left (\frac {8 x^{3} a}{27}-\frac {2}{3}\right ) a}{x^{2}}+\frac {2 \left (\frac {2}{27} a \,x^{4}-\frac {2}{3} x \right ) a}{x^{3}}-\left (\frac {2}{27} a \,x^{4}-\frac {2}{3} x \right ) a^{2}\right )}^{3} x^{8}}{27 a^{4} \left (\frac {2}{27} a \,x^{4}-\frac {2}{3} x \right )^{5}} \end {align*}
Since the Abel invariant depends on \(x\) then unable to solve this ode at this time.
Unable to complete the solution now.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} y^{\prime }+a y^{3}-y^{2} a \,x^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-a y^{3}+y^{2} a \,x^{2}}{x^{2}} \end {array} \]
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 trying Abel <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 148
dsolve(x^2*diff(y(x),x) + a*y(x)^3 - a*x^2*y(x)^2=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {1}{-a x -2^{\frac {2}{3}} \left (-a \right )^{\frac {2}{3}} \operatorname {RootOf}\left (\operatorname {AiryBi}\left (\frac {\left (\textit {\_Z}^{2} 2^{\frac {1}{3}} \left (-a \right )^{\frac {1}{3}} x -1\right ) 2^{\frac {2}{3}}}{2 \left (-a \right )^{\frac {1}{3}} x}\right ) c_{1} \textit {\_Z} +\textit {\_Z} \operatorname {AiryAi}\left (\frac {\left (\textit {\_Z}^{2} 2^{\frac {1}{3}} \left (-a \right )^{\frac {1}{3}} x -1\right ) 2^{\frac {2}{3}}}{2 \left (-a \right )^{\frac {1}{3}} x}\right )+\operatorname {AiryBi}\left (1, \frac {\left (\textit {\_Z}^{2} 2^{\frac {1}{3}} \left (-a \right )^{\frac {1}{3}} x -1\right ) 2^{\frac {2}{3}}}{2 \left (-a \right )^{\frac {1}{3}} x}\right ) c_{1} +\operatorname {AiryAi}\left (1, \frac {\left (\textit {\_Z}^{2} 2^{\frac {1}{3}} \left (-a \right )^{\frac {1}{3}} x -1\right ) 2^{\frac {2}{3}}}{2 \left (-a \right )^{\frac {1}{3}} x}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.436 (sec). Leaf size: 267
DSolve[x^2*y'[x] + a*y[x]^3 - a*x^2*y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\left (-\frac {1}{2^{2/3} a^{2/3} y(x)}-\frac {\sqrt [3]{a} x}{2^{2/3}}\right ) \operatorname {AiryAi}\left (\left (-\frac {\sqrt [3]{a} x}{2^{2/3}}-\frac {1}{2^{2/3} a^{2/3} y(x)}\right )^2+\frac {1}{\sqrt [3]{2} \sqrt [3]{a} x}\right )+\operatorname {AiryAiPrime}\left (\left (-\frac {\sqrt [3]{a} x}{2^{2/3}}-\frac {1}{2^{2/3} a^{2/3} y(x)}\right )^2+\frac {1}{\sqrt [3]{2} \sqrt [3]{a} x}\right )}{\left (-\frac {1}{2^{2/3} a^{2/3} y(x)}-\frac {\sqrt [3]{a} x}{2^{2/3}}\right ) \operatorname {AiryBi}\left (\left (-\frac {\sqrt [3]{a} x}{2^{2/3}}-\frac {1}{2^{2/3} a^{2/3} y(x)}\right )^2+\frac {1}{\sqrt [3]{2} \sqrt [3]{a} x}\right )+\operatorname {AiryBiPrime}\left (\left (-\frac {\sqrt [3]{a} x}{2^{2/3}}-\frac {1}{2^{2/3} a^{2/3} y(x)}\right )^2+\frac {1}{\sqrt [3]{2} \sqrt [3]{a} x}\right )}+c_1=0,y(x)\right ] \]