Internal problem ID [10752]
Internal file name [OUTPUT/9700_Monday_June_06_2022_03_29_14_PM_62636486/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 1.3.3-2. Equations
of the form \(y y'=f_1(x) y+f_0(x)\)
Problem number: 16.
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.
\[ \boxed {y y^{\prime }-a \left (1-\frac {b}{\sqrt {x}}\right ) y=\frac {a^{2} b}{\sqrt {x}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime } \sqrt {x}-\sqrt {x}\, y a +a y b -a^{2} b =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\sqrt {x}\, y a -a y b +a^{2} b}{y \sqrt {x}} \end {array} \]
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: 269
dsolve(y(x)*diff(y(x),x)-a*(1-b*x^(-1/2))*y(x)=a^2*b*x^(-1/2),y(x), singsol=all)
\[ \frac {\left (b^{2}\right )^{\frac {1}{3}} c_{1} 2^{\frac {2}{3}} \left (-\sqrt {x}+b \right ) \operatorname {AiryBi}\left (\frac {2^{\frac {1}{3}} \left (-2 \sqrt {x}\, a b +\left (b^{2}+x \right ) a -y \left (x \right )\right )}{2 \left (b^{2}\right )^{\frac {1}{3}} a}\right )+2 \operatorname {AiryBi}\left (1, \frac {2^{\frac {1}{3}} \left (-2 \sqrt {x}\, a b +\left (b^{2}+x \right ) a -y \left (x \right )\right )}{2 \left (b^{2}\right )^{\frac {1}{3}} a}\right ) c_{1} b -2 \operatorname {AiryAi}\left (1, \frac {2^{\frac {1}{3}} \left (-2 \sqrt {x}\, a b +\left (b^{2}+x \right ) a -y \left (x \right )\right )}{2 \left (b^{2}\right )^{\frac {1}{3}} a}\right ) b -\left (b^{2}\right )^{\frac {1}{3}} 2^{\frac {2}{3}} \left (-\sqrt {x}+b \right ) \operatorname {AiryAi}\left (\frac {2^{\frac {1}{3}} \left (-2 \sqrt {x}\, a b +\left (b^{2}+x \right ) a -y \left (x \right )\right )}{2 \left (b^{2}\right )^{\frac {1}{3}} a}\right )}{\left (b^{2}\right )^{\frac {1}{3}} 2^{\frac {2}{3}} \left (-\sqrt {x}+b \right ) \operatorname {AiryBi}\left (\frac {2^{\frac {1}{3}} \left (-2 \sqrt {x}\, a b +\left (b^{2}+x \right ) a -y \left (x \right )\right )}{2 \left (b^{2}\right )^{\frac {1}{3}} a}\right )+2 \operatorname {AiryBi}\left (1, \frac {2^{\frac {1}{3}} \left (-2 \sqrt {x}\, a b +\left (b^{2}+x \right ) a -y \left (x \right )\right )}{2 \left (b^{2}\right )^{\frac {1}{3}} a}\right ) b} = 0 \]
✓ Solution by Mathematica
Time used: 1.905 (sec). Leaf size: 323
DSolve[y[x]*y'[x]-a*(1-b*x^(-1/2))*y[x]==a^2*b*x^(-1/2),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\sqrt [3]{-1} 2^{2/3} \sqrt [3]{\left (b-\sqrt {x}\right )^3} \operatorname {AiryAi}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} \left (\left (b-\sqrt {x}\right )^3\right )^{2/3} \left (a \left (b-\sqrt {x}\right )^2-y(x)\right )}{a b^{2/3} \left (b-\sqrt {x}\right )^2}\right )-2 \sqrt [3]{b} \operatorname {AiryAiPrime}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} \left (\left (b-\sqrt {x}\right )^3\right )^{2/3} \left (a \left (b-\sqrt {x}\right )^2-y(x)\right )}{a b^{2/3} \left (b-\sqrt {x}\right )^2}\right )}{\sqrt [3]{-1} 2^{2/3} \sqrt [3]{\left (b-\sqrt {x}\right )^3} \operatorname {AiryBi}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} \left (\left (b-\sqrt {x}\right )^3\right )^{2/3} \left (a \left (b-\sqrt {x}\right )^2-y(x)\right )}{a b^{2/3} \left (b-\sqrt {x}\right )^2}\right )-2 \sqrt [3]{b} \operatorname {AiryBiPrime}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} \left (\left (b-\sqrt {x}\right )^3\right )^{2/3} \left (a \left (b-\sqrt {x}\right )^2-y(x)\right )}{a b^{2/3} \left (b-\sqrt {x}\right )^2}\right )}+c_1=0,y(x)\right ] \]