Internal problem ID [10725]
Internal file name [OUTPUT/9673_Monday_June_06_2022_03_21_32_PM_48237524/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: 1.
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 A`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }-\left (a x +b \right ) y=1} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-\left (a x +b \right ) y=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\left (a x +b \right ) y+1}{y} \end {array} \]
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: 215
dsolve(y(x)*diff(y(x),x)=(a*x+b)*y(x)+1,y(x), singsol=all)
\[ \frac {\left (a x +b \right ) \left (\operatorname {AiryBi}\left (-\frac {\left (-2 a y \left (x \right )+\left (a x +b \right )^{2}\right ) 2^{\frac {2}{3}}}{4 \left (-a^{2}\right )^{\frac {1}{3}}}\right ) c_{1} -\operatorname {AiryAi}\left (-\frac {\left (-2 a y \left (x \right )+\left (a x +b \right )^{2}\right ) 2^{\frac {2}{3}}}{4 \left (-a^{2}\right )^{\frac {1}{3}}}\right )\right ) \left (-a^{2}\right )^{\frac {1}{3}} 2^{\frac {1}{3}}+2 \left (\operatorname {AiryBi}\left (1, -\frac {\left (-2 a y \left (x \right )+\left (a x +b \right )^{2}\right ) 2^{\frac {2}{3}}}{4 \left (-a^{2}\right )^{\frac {1}{3}}}\right ) c_{1} -\operatorname {AiryAi}\left (1, -\frac {\left (-2 a y \left (x \right )+\left (a x +b \right )^{2}\right ) 2^{\frac {2}{3}}}{4 \left (-a^{2}\right )^{\frac {1}{3}}}\right )\right ) a}{2^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {1}{3}} \left (a x +b \right ) \operatorname {AiryBi}\left (-\frac {\left (-2 a y \left (x \right )+\left (a x +b \right )^{2}\right ) 2^{\frac {2}{3}}}{4 \left (-a^{2}\right )^{\frac {1}{3}}}\right )+2 \operatorname {AiryBi}\left (1, -\frac {\left (-2 a y \left (x \right )+\left (a x +b \right )^{2}\right ) 2^{\frac {2}{3}}}{4 \left (-a^{2}\right )^{\frac {1}{3}}}\right ) a} = 0 \]
✓ Solution by Mathematica
Time used: 0.901 (sec). Leaf size: 161
DSolve[y[x]*y'[x]==(a*x+b)*y[x]+1,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\sqrt [3]{2} (a x+b) \operatorname {AiryAi}\left (\frac {(b+a x)^2-2 a y(x)}{2 \sqrt [3]{2} a^{2/3}}\right )-2 \sqrt [3]{a} \operatorname {AiryAiPrime}\left (\frac {(b+a x)^2-2 a y(x)}{2 \sqrt [3]{2} a^{2/3}}\right )}{\sqrt [3]{2} (a x+b) \operatorname {AiryBi}\left (\frac {(b+a x)^2-2 a y(x)}{2 \sqrt [3]{2} a^{2/3}}\right )-2 \sqrt [3]{a} \operatorname {AiryBiPrime}\left (\frac {(b+a x)^2-2 a y(x)}{2 \sqrt [3]{2} a^{2/3}}\right )}+c_1=0,y(x)\right ] \]