problem 79

Internal problem ID [3343]
Internal ﬁle name [OUTPUT/2835_Sunday_June_05_2022_08_41_23_AM_17871836/index.tex]

Book: Ordinary diﬀerential equations and their solutions. By George Moseley Murphy. 1960
Section: Various 3
Problem number: 79.
ODE order: 1.
ODE degree: 1.

3.25.1 Solving as abelFirstKind ode

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 }&=-3 a y^{3}-6 y^{2} a x\tag {1} \end {align*}

Therefore \begin {align*} f_0(x) &= 0\\ f_1(x) &= 0\\ f_2(x) &= -6 a x\\ f_3(x) &= -3 a \end {align*}

Since \(f_2(x)=-6 a x\) 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 {-6 a x}{-9 a} \right ) \\ &= u \left (x \right )-\frac {2 x}{3} \end {align*}

The above transformation applied to (1) gives a new ODE as \begin {align*} u^{\prime }\left (x \right ) = \frac {2}{3}+4 a u \left (x \right ) x^{2}-\frac {16 a \,x^{3}}{9}-3 a 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 {2}{3}+4 a u \left (x \right ) x^{2}-\frac {16 a \,x^{3}}{9}-3 a u \left (x \right )^{3}\tag {1} \end {align*}

Therefore \begin {align*} f_0(x) &= \frac {2}{3}-\frac {16 a \,x^{3}}{9}\\ f_1(x) &= 4 a \,x^{2}\\ f_2(x) &= 0\\ f_3(x) &= -3 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 (-16 a^{2} x^{2}-36 \left (\frac {2}{3}-\frac {16 a \,x^{3}}{9}\right ) a^{2} x^{2}\right )^{3}}{2187 a^{4} \left (\frac {2}{3}-\frac {16 a \,x^{3}}{9}\right )^{5}} \end {align*}

Since the Abel invariant depends on \(x\) then unable to solve this ode at this time.

3.25.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+3 a \left (2 x +y\right ) y^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-3 a \left (2 x +y\right ) y^{2} \end {array} \]

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`

\[ y \left (x \right ) = \frac {1}{3 a \,x^{2}+\operatorname {RootOf}\left (3^{\frac {1}{3}} \left (-a \right )^{\frac {1}{3}} \operatorname {AiryBi}\left (\textit {\_Z} \right ) c_{1} x +3^{\frac {1}{3}} \left (-a \right )^{\frac {1}{3}} x \operatorname {AiryAi}\left (\textit {\_Z} \right )+\operatorname {AiryBi}\left (1, \textit {\_Z}\right ) c_{1} +\operatorname {AiryAi}\left (1, \textit {\_Z}\right )\right ) 3^{\frac {1}{3}} \left (-a \right )^{\frac {1}{3}}} \]

\[ \text {Solve}\left [\frac {\sqrt [3]{-3} \sqrt [3]{a} x \operatorname {AiryAi}\left ((-3)^{2/3} a^{2/3} x^2-\frac {(-1)^{2/3}}{\sqrt [3]{3} \sqrt [3]{a} y(x)}\right )+\operatorname {AiryAiPrime}\left ((-3)^{2/3} a^{2/3} x^2-\frac {(-1)^{2/3}}{\sqrt [3]{3} \sqrt [3]{a} y(x)}\right )}{\sqrt [3]{-3} \sqrt [3]{a} x \operatorname {AiryBi}\left ((-3)^{2/3} a^{2/3} x^2-\frac {(-1)^{2/3}}{\sqrt [3]{3} \sqrt [3]{a} y(x)}\right )+\operatorname {AiryBiPrime}\left ((-3)^{2/3} a^{2/3} x^2-\frac {(-1)^{2/3}}{\sqrt [3]{3} \sqrt [3]{a} y(x)}\right )}+c_1=0,y(x)\right ] \]

3.25 problem 79

3.25.1 Solving as abelFirstKind ode

3.25.2 Maple step by step solution