Internal problem ID [3341]
Internal file name [OUTPUT/2833_Sunday_June_05_2022_08_41_21_AM_36827218/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 3
Problem number: 77.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "abelFirstKind"
Maple gives the following as the ode type
[_Abel]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }+\left (a x +y\right ) y^{2}=0} \]
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 }&=-y^{3}-y^{2} a x\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= 0\\ f_1(x) &= 0\\ f_2(x) &= -a x\\ f_3(x) &= -1 \end {align*}
Since \(f_2(x)=-a x\) 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 x}{-3} \right ) \\ &= u \left (x \right )-\frac {a x}{3} \end {align*}
The above transformation applied to (1) gives a new ODE as \begin {align*} u^{\prime }\left (x \right ) = \frac {a}{3}+\frac {a^{2} x^{2} u \left (x \right )}{3}-\frac {2 a^{3} x^{3}}{27}-u \left (x \right )^{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}{3}+\frac {a^{2} x^{2} u \left (x \right )}{3}-\frac {2 a^{3} x^{3}}{27}-u \left (x \right )^{3}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= \frac {1}{3} a -\frac {2}{27} a^{3} x^{3}\\ f_1(x) &= \frac {a^{2} x^{2}}{3}\\ f_2(x) &= 0\\ f_3(x) &= -1 \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 {2 a^{3} x^{2}}{9}-\left (\frac {1}{3} a -\frac {2}{27} a^{3} x^{3}\right ) a^{2} x^{2}\right )^{3}}{27 \left (\frac {1}{3} a -\frac {2}{27} a^{3} x^{3}\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.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+\left (a x +y\right ) y^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\left (a x +y\right ) y^{2} \end {array} \]
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: 71
dsolve(diff(y(x),x)+(a*x+y(x))*y(x)^2=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {2 a}{x^{2} a^{2}+2 \operatorname {RootOf}\left (2^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {1}{3}} \operatorname {AiryBi}\left (\textit {\_Z} \right ) c_{1} x +2^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {1}{3}} x \operatorname {AiryAi}\left (\textit {\_Z} \right )+2 \operatorname {AiryBi}\left (1, \textit {\_Z}\right ) c_{1} +2 \operatorname {AiryAi}\left (1, \textit {\_Z}\right )\right ) 2^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {1}{3}}} \]
✓ Solution by Mathematica
Time used: 0.247 (sec). Leaf size: 195
DSolve[y'[x]+(a x+y[x])y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\operatorname {AiryAiPrime}\left (\frac {\sqrt [3]{-\frac {1}{2}} \sqrt [3]{a}}{y(x)}-\frac {1}{2} \sqrt [3]{-\frac {1}{2}} a^{4/3} x^2\right )-\left (-\frac {1}{2}\right )^{2/3} a^{2/3} x \operatorname {AiryAi}\left (\frac {\sqrt [3]{-\frac {1}{2}} \sqrt [3]{a}}{y(x)}-\frac {1}{2} \sqrt [3]{-\frac {1}{2}} a^{4/3} x^2\right )}{\operatorname {AiryBiPrime}\left (\frac {\sqrt [3]{-\frac {1}{2}} \sqrt [3]{a}}{y(x)}-\frac {1}{2} \sqrt [3]{-\frac {1}{2}} a^{4/3} x^2\right )-\left (-\frac {1}{2}\right )^{2/3} a^{2/3} x \operatorname {AiryBi}\left (\frac {\sqrt [3]{-\frac {1}{2}} \sqrt [3]{a}}{y(x)}-\frac {1}{2} \sqrt [3]{-\frac {1}{2}} a^{4/3} x^2\right )}+c_1=0,y(x)\right ] \]