Internal problem ID [8483]
Internal file name [OUTPUT/7416_Sunday_June_05_2022_10_54_46_PM_71056415/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 147.
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.
\[ \boxed {x^{2} y^{\prime }+a \,x^{2} y^{3}+b y^{2}=0} \]
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 }&=-a y^{3}-\frac {b y^{2}}{x^{2}}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= 0\\ f_1(x) &= 0\\ f_2(x) &= -\frac {b}{x^{2}}\\ f_3(x) &= -a \end {align*}
Since \(f_2(x)=-\frac {b}{x^{2}}\) 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 {-\frac {b}{x^{2}}}{-3 a} \right ) \\ &= u \left (x \right )-\frac {b}{3 x^{2} a} \end {align*}
The above transformation applied to (1) gives a new ODE as \begin {align*} u^{\prime }\left (x \right ) = -a u \left (x \right )^{3}+\frac {u \left (x \right ) b^{2}}{3 x^{4} a}-\frac {2 b}{3 x^{3} a}-\frac {2 b^{3}}{27 x^{6} a^{2}}\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 )&=-a u \left (x \right )^{3}+\frac {u \left (x \right ) b^{2}}{3 x^{4} a}-\frac {18 a \,x^{3} b +2 b^{3}}{27 a^{2} x^{6}}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= -\frac {2 b}{3 x^{3} a}-\frac {2 b^{3}}{27 x^{6} a^{2}}\\ f_1(x) &= \frac {b^{2}}{3 x^{4} a}\\ f_2(x) &= 0\\ f_3(x) &= -a \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 (\left (\frac {2 b}{a \,x^{4}}+\frac {4 b^{3}}{9 a^{2} x^{7}}\right ) a -\frac {\left (-\frac {2 b}{3 x^{3} a}-\frac {2 b^{3}}{27 x^{6} a^{2}}\right ) b^{2}}{x^{4}}\right )}^{3}}{27 a^{4} \left (-\frac {2 b}{3 x^{3} a}-\frac {2 b^{3}}{27 x^{6} a^{2}}\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.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} y^{\prime }+a \,x^{2} y^{3}+b y^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {a \,x^{2} y^{3}+b y^{2}}{x^{2}} \end {array} \]
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: 182
dsolve(x^2*diff(y(x),x) + a*x^2*y(x)^3 + b*y(x)^2=0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {2^{\frac {1}{3}} a b x}{2^{\frac {1}{3}} a \,b^{2}-2 \left (b^{2} a^{2}\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\operatorname {AiryBi}\left (\frac {-a 2^{\frac {2}{3}} x +2 \textit {\_Z}^{2} \left (b^{2} a^{2}\right )^{\frac {1}{3}}}{2 \left (b^{2} a^{2}\right )^{\frac {1}{3}}}\right ) c_{1} \textit {\_Z} +\textit {\_Z} \operatorname {AiryAi}\left (\frac {-a 2^{\frac {2}{3}} x +2 \textit {\_Z}^{2} \left (b^{2} a^{2}\right )^{\frac {1}{3}}}{2 \left (b^{2} a^{2}\right )^{\frac {1}{3}}}\right )+\operatorname {AiryBi}\left (1, \frac {-a 2^{\frac {2}{3}} x +2 \textit {\_Z}^{2} \left (b^{2} a^{2}\right )^{\frac {1}{3}}}{2 \left (b^{2} a^{2}\right )^{\frac {1}{3}}}\right ) c_{1} +\operatorname {AiryAi}\left (1, \frac {-a 2^{\frac {2}{3}} x +2 \textit {\_Z}^{2} \left (b^{2} a^{2}\right )^{\frac {1}{3}}}{2 \left (b^{2} a^{2}\right )^{\frac {1}{3}}}\right )\right ) x} \]
✓ Solution by Mathematica
Time used: 0.581 (sec). Leaf size: 343
DSolve[x^2*y'[x] + a*x^2*y[x]^3 + b*y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\left (\frac {b^{2/3}}{2^{2/3} \sqrt [3]{a} x}+\frac {1}{2^{2/3} \sqrt [3]{a} \sqrt [3]{b} y(x)}\right ) \operatorname {AiryAi}\left (\left (\frac {b^{2/3}}{2^{2/3} \sqrt [3]{a} x}+\frac {1}{2^{2/3} \sqrt [3]{a} y(x) \sqrt [3]{b}}\right )^2-\frac {\sqrt [3]{a} x}{\sqrt [3]{2} b^{2/3}}\right )+\operatorname {AiryAiPrime}\left (\left (\frac {b^{2/3}}{2^{2/3} \sqrt [3]{a} x}+\frac {1}{2^{2/3} \sqrt [3]{a} y(x) \sqrt [3]{b}}\right )^2-\frac {\sqrt [3]{a} x}{\sqrt [3]{2} b^{2/3}}\right )}{\left (\frac {b^{2/3}}{2^{2/3} \sqrt [3]{a} x}+\frac {1}{2^{2/3} \sqrt [3]{a} \sqrt [3]{b} y(x)}\right ) \operatorname {AiryBi}\left (\left (\frac {b^{2/3}}{2^{2/3} \sqrt [3]{a} x}+\frac {1}{2^{2/3} \sqrt [3]{a} y(x) \sqrt [3]{b}}\right )^2-\frac {\sqrt [3]{a} x}{\sqrt [3]{2} b^{2/3}}\right )+\operatorname {AiryBiPrime}\left (\left (\frac {b^{2/3}}{2^{2/3} \sqrt [3]{a} x}+\frac {1}{2^{2/3} \sqrt [3]{a} y(x) \sqrt [3]{b}}\right )^2-\frac {\sqrt [3]{a} x}{\sqrt [3]{2} b^{2/3}}\right )}+c_1=0,y(x)\right ] \]