Internal problem ID [8545]
Internal file name [OUTPUT/7478_Sunday_June_05_2022_10_57_36_PM_52681442/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 209.
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]
\[ \boxed {y y^{\prime }-\sqrt {a y^{2}+b}=0} \]
Integrating both sides gives \begin{align*} \int \frac {y}{\sqrt {a \,y^{2}+b}}d y &= \int d x \\ \frac {\sqrt {a y^{2}+b}}{a}&=x +c_{1} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} \frac {\sqrt {a y^{2}+b}}{a} &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ \frac {\sqrt {a y^{2}+b}}{a} = x +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-\sqrt {a y^{2}+b}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\sqrt {a y^{2}+b}}{y} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y y^{\prime }}{\sqrt {a y^{2}+b}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y y^{\prime }}{\sqrt {a y^{2}+b}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\sqrt {a y^{2}+b}}{a}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\frac {\sqrt {a \left (c_{1}^{2} a^{2}+2 c_{1} a^{2} x +a^{2} x^{2}-b \right )}}{a}, y=-\frac {\sqrt {a \left (c_{1}^{2} a^{2}+2 c_{1} a^{2} x +a^{2} x^{2}-b \right )}}{a}\right \} \end {array} \]
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: 25
dsolve(y(x)*diff(y(x),x)-sqrt(a*y(x)^2+b)=0,y(x), singsol=all)
\[ \frac {-\sqrt {a y \left (x \right )^{2}+b}+\left (x +c_{1} \right ) a}{a} = 0 \]
✓ Solution by Mathematica
Time used: 0.531 (sec). Leaf size: 94
DSolve[y[x]*y'[x]-Sqrt[a*y[x]^2+b]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {-b+a^2 (x+c_1){}^2}}{\sqrt {a}} \\ y(x)\to \frac {\sqrt {-b+a^2 (x+c_1){}^2}}{\sqrt {a}} \\ y(x)\to -\frac {i \sqrt {b}}{\sqrt {a}} \\ y(x)\to \frac {i \sqrt {b}}{\sqrt {a}} \\ \end{align*}