Internal problem ID [10726]
Internal file name [OUTPUT/9674_Monday_June_06_2022_03_21_33_PM_94051857/index.tex]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.3. Abel Equations of the Second Kind. subsection 1.3.2. Equations of
the form \(y y'=f(x) y+1\)
Problem number: 2.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational, [_Abel, `2nd type`, `class B`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }-\frac {y}{\left (a x +b \right )^{2}}=1} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime } a^{2} x^{2}+2 y y^{\prime } a b x +y y^{\prime } b^{2}-a^{2} x^{2}-2 a b x -b^{2}-y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {a^{2} x^{2}+2 a b x +b^{2}+y}{y a^{2} x^{2}+2 y a b x +y b^{2}} \end {array} \]
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: 557
dsolve(y(x)*diff(y(x),x)=(a*x+b)^(-2)*y(x)+1,y(x), singsol=all)
\[ \frac {-a \left (-\operatorname {AiryBi}\left (-\frac {2^{\frac {2}{3}} \left (-\frac {a^{2} \left (a x +b \right )^{2} y \left (x \right )^{2}}{2}+\left (-a^{2} x -a b \right ) y \left (x \right )+a^{4} x^{3}+3 a^{3} b \,x^{2}+3 a^{2} b^{2} x +a \,b^{3}-\frac {1}{2}\right )}{2 \left (a^{2}\right )^{\frac {1}{3}} \left (a x +b \right )^{2}}\right ) c_{1} +\operatorname {AiryAi}\left (-\frac {2^{\frac {2}{3}} \left (-\frac {a^{2} \left (a x +b \right )^{2} y \left (x \right )^{2}}{2}+\left (-a^{2} x -a b \right ) y \left (x \right )+a^{4} x^{3}+3 a^{3} b \,x^{2}+3 a^{2} b^{2} x +a \,b^{3}-\frac {1}{2}\right )}{2 \left (a^{2}\right )^{\frac {1}{3}} \left (a x +b \right )^{2}}\right )\right ) \left (1+a \left (a x +b \right ) y \left (x \right )\right ) 2^{\frac {1}{3}}+2 \left (a x +b \right ) \left (a^{2}\right )^{\frac {2}{3}} \left (\operatorname {AiryBi}\left (1, -\frac {2^{\frac {2}{3}} \left (-\frac {a^{2} \left (a x +b \right )^{2} y \left (x \right )^{2}}{2}+\left (-a^{2} x -a b \right ) y \left (x \right )+a^{4} x^{3}+3 a^{3} b \,x^{2}+3 a^{2} b^{2} x +a \,b^{3}-\frac {1}{2}\right )}{2 \left (a^{2}\right )^{\frac {1}{3}} \left (a x +b \right )^{2}}\right ) c_{1} -\operatorname {AiryAi}\left (1, -\frac {2^{\frac {2}{3}} \left (-\frac {a^{2} \left (a x +b \right )^{2} y \left (x \right )^{2}}{2}+\left (-a^{2} x -a b \right ) y \left (x \right )+a^{4} x^{3}+3 a^{3} b \,x^{2}+3 a^{2} b^{2} x +a \,b^{3}-\frac {1}{2}\right )}{2 \left (a^{2}\right )^{\frac {1}{3}} \left (a x +b \right )^{2}}\right )\right )}{a 2^{\frac {1}{3}} \left (1+a \left (a x +b \right ) y \left (x \right )\right ) \operatorname {AiryBi}\left (-\frac {2^{\frac {2}{3}} \left (-\frac {a^{2} \left (a x +b \right )^{2} y \left (x \right )^{2}}{2}+\left (-a^{2} x -a b \right ) y \left (x \right )+a^{4} x^{3}+3 a^{3} b \,x^{2}+3 a^{2} b^{2} x +a \,b^{3}-\frac {1}{2}\right )}{2 \left (a^{2}\right )^{\frac {1}{3}} \left (a x +b \right )^{2}}\right )+2 \operatorname {AiryBi}\left (1, -\frac {2^{\frac {2}{3}} \left (-\frac {a^{2} \left (a x +b \right )^{2} y \left (x \right )^{2}}{2}+\left (-a^{2} x -a b \right ) y \left (x \right )+a^{4} x^{3}+3 a^{3} b \,x^{2}+3 a^{2} b^{2} x +a \,b^{3}-\frac {1}{2}\right )}{2 \left (a^{2}\right )^{\frac {1}{3}} \left (a x +b \right )^{2}}\right ) \left (a^{2}\right )^{\frac {2}{3}} \left (a x +b \right )} = 0 \]
✓ Solution by Mathematica
Time used: 2.233 (sec). Leaf size: 561
DSolve[y[x]*y'[x]==(a*x+b)^(-2)*y[x]+1,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {a y(x) (a x+b) \operatorname {AiryAi}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )+\operatorname {AiryAi}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )+2^{2/3} \sqrt [3]{a (a x+b)^3} \operatorname {AiryAiPrime}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )}{a y(x) (a x+b) \operatorname {AiryBi}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )+\operatorname {AiryBi}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )+2^{2/3} \sqrt [3]{a (a x+b)^3} \operatorname {AiryBiPrime}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )}+c_1=0,y(x)\right ] \]