Internal problem ID [3529]
Internal file name [OUTPUT/3022_Sunday_June_05_2022_08_49_52_AM_81239861/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 10
Problem number: 273.
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.
\[ \boxed {y^{\prime } x^{2}+a y^{2}+y^{3} b \,x^{2}=0} \]
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 }&=-b y^{3}-\frac {a y^{2}}{x^{2}}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= 0\\ f_1(x) &= 0\\ f_2(x) &= -\frac {a}{x^{2}}\\ f_3(x) &= -b \end {align*}
Since \(f_2(x)=-\frac {a}{x^{2}}\) 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 {a}{x^{2}}}{-3 b} \right ) \\ &= u \left (x \right )-\frac {a}{3 b \,x^{2}} \end {align*}
The above transformation applied to (1) gives a new ODE as \begin {align*} u^{\prime }\left (x \right ) = -b u \left (x \right )^{3}+\frac {u \left (x \right ) a^{2}}{3 b \,x^{4}}-\frac {2 a}{3 b \,x^{3}}-\frac {2 a^{3}}{27 b^{2} x^{6}}\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 )&=-b u \left (x \right )^{3}+\frac {u \left (x \right ) a^{2}}{3 b \,x^{4}}-\frac {18 b \,x^{3} a +2 a^{3}}{27 x^{6} b^{2}}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= -\frac {2 a}{3 b \,x^{3}}-\frac {2 a^{3}}{27 b^{2} x^{6}}\\ f_1(x) &= \frac {a^{2}}{3 b \,x^{4}}\\ f_2(x) &= 0\\ f_3(x) &= -b \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 a}{x^{4} b}+\frac {4 a^{3}}{9 x^{7} b^{2}}\right ) b -\frac {\left (-\frac {2 a}{3 b \,x^{3}}-\frac {2 a^{3}}{27 b^{2} x^{6}}\right ) a^{2}}{x^{4}}\right )}^{3}}{27 b^{4} \left (-\frac {2 a}{3 b \,x^{3}}-\frac {2 a^{3}}{27 b^{2} x^{6}}\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.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } x^{2}+a y^{2}+y^{3} b \,x^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {a y^{2}+y^{3} b \,x^{2}}{x^{2}} \end {array} \]
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: 178
dsolve(x^2*diff(y(x),x)+a*y(x)^2+b*x^2*y(x)^3 = 0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {2^{\frac {1}{3}} a b x}{2^{\frac {1}{3}} a^{2} b -2 \left (a^{2} b^{2}\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\operatorname {AiryBi}\left (-\frac {b 2^{\frac {2}{3}} x -2 \textit {\_Z}^{2} \left (a^{2} b^{2}\right )^{\frac {1}{3}}}{2 \left (a^{2} b^{2}\right )^{\frac {1}{3}}}\right ) c_{1} \textit {\_Z} +\textit {\_Z} \operatorname {AiryAi}\left (-\frac {b 2^{\frac {2}{3}} x -2 \textit {\_Z}^{2} \left (a^{2} b^{2}\right )^{\frac {1}{3}}}{2 \left (a^{2} b^{2}\right )^{\frac {1}{3}}}\right )+\operatorname {AiryBi}\left (1, -\frac {b 2^{\frac {2}{3}} x -2 \textit {\_Z}^{2} \left (a^{2} b^{2}\right )^{\frac {1}{3}}}{2 \left (a^{2} b^{2}\right )^{\frac {1}{3}}}\right ) c_{1} +\operatorname {AiryAi}\left (1, -\frac {b 2^{\frac {2}{3}} x -2 \textit {\_Z}^{2} \left (a^{2} b^{2}\right )^{\frac {1}{3}}}{2 \left (a^{2} b^{2}\right )^{\frac {1}{3}}}\right )\right ) x} \]
✓ Solution by Mathematica
Time used: 0.609 (sec). Leaf size: 343
DSolve[x^2 y'[x]+a y[x]^2+b x^2 y[x]^3==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\left (\frac {a^{2/3}}{2^{2/3} \sqrt [3]{b} x}+\frac {1}{2^{2/3} \sqrt [3]{a} \sqrt [3]{b} y(x)}\right ) \operatorname {AiryAi}\left (\left (\frac {a^{2/3}}{2^{2/3} \sqrt [3]{b} x}+\frac {1}{2^{2/3} \sqrt [3]{b} y(x) \sqrt [3]{a}}\right )^2-\frac {\sqrt [3]{b} x}{\sqrt [3]{2} a^{2/3}}\right )+\operatorname {AiryAiPrime}\left (\left (\frac {a^{2/3}}{2^{2/3} \sqrt [3]{b} x}+\frac {1}{2^{2/3} \sqrt [3]{b} y(x) \sqrt [3]{a}}\right )^2-\frac {\sqrt [3]{b} x}{\sqrt [3]{2} a^{2/3}}\right )}{\left (\frac {a^{2/3}}{2^{2/3} \sqrt [3]{b} x}+\frac {1}{2^{2/3} \sqrt [3]{a} \sqrt [3]{b} y(x)}\right ) \operatorname {AiryBi}\left (\left (\frac {a^{2/3}}{2^{2/3} \sqrt [3]{b} x}+\frac {1}{2^{2/3} \sqrt [3]{b} y(x) \sqrt [3]{a}}\right )^2-\frac {\sqrt [3]{b} x}{\sqrt [3]{2} a^{2/3}}\right )+\operatorname {AiryBiPrime}\left (\left (\frac {a^{2/3}}{2^{2/3} \sqrt [3]{b} x}+\frac {1}{2^{2/3} \sqrt [3]{b} y(x) \sqrt [3]{a}}\right )^2-\frac {\sqrt [3]{b} x}{\sqrt [3]{2} a^{2/3}}\right )}+c_1=0,y(x)\right ] \]