Internal problem ID [8360]
Internal file name [OUTPUT/7293_Sunday_June_05_2022_05_43_04_PM_74664346/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 23.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {y^{\prime }+a y^{2}=b} \]
Integrating both sides gives \begin {align*} \int \frac {1}{-a \,y^{2}+b}d y &= x +c_{1}\\ \frac {\operatorname {arctanh}\left (\frac {a y}{\sqrt {a b}}\right )}{\sqrt {a b}}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {\tanh \left (c_{1} \sqrt {a b}+x \sqrt {a b}\right ) \sqrt {a b}}{a} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\tanh \left (c_{1} \sqrt {a b}+x \sqrt {a b}\right ) \sqrt {a b}}{a} \\ \end{align*}
Verification of solutions
\[ y = \frac {\tanh \left (c_{1} \sqrt {a b}+x \sqrt {a b}\right ) \sqrt {a b}}{a} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+a y^{2}=b \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-a y^{2}+b \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-a y^{2}+b}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-a y^{2}+b}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\mathrm {arctanh}\left (\frac {a y}{\sqrt {a b}}\right )}{\sqrt {a b}}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\tanh \left (c_{1} \sqrt {a b}+x \sqrt {a b}\right ) \sqrt {a b}}{a} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 23
dsolve(diff(y(x),x) + a*y(x)^2 - b=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\tanh \left (\sqrt {a b}\, \left (x +c_{1} \right )\right ) \sqrt {a b}}{a} \]
✓ Solution by Mathematica
Time used: 5.188 (sec). Leaf size: 63
DSolve[y'[x] + a*y[x]^2 - b==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {b} \tanh \left (\sqrt {a} \sqrt {b} (x+c_1)\right )}{\sqrt {a}} \\ y(x)\to -\frac {\sqrt {b}}{\sqrt {a}} \\ y(x)\to \frac {\sqrt {b}}{\sqrt {a}} \\ \end{align*}