Internal problem ID [7090]
Internal file name [OUTPUT/6076_Sunday_June_05_2022_04_18_45_PM_61799293/index.tex]
Book: Own collection of miscellaneous problems
Section: section 1.0
Problem number: 46.
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]
\[ \boxed {f^{\prime }-\frac {1}{f}=0} \]
Integrating both sides gives \begin {align*} \int f d f &= x +c_{1}\\ \frac {f^{2}}{2}&=x +c_{1} \end {align*}
Solving for \(f\) gives these solutions \begin {align*} f_1&=\sqrt {2 c_{1} +2 x}\\ f_2&=-\sqrt {2 c_{1} +2 x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} f &= \sqrt {2 c_{1} +2 x} \\ \tag{2} f &= -\sqrt {2 c_{1} +2 x} \\ \end{align*}
Verification of solutions
\[ f = \sqrt {2 c_{1} +2 x} \] Verified OK.
\[ f = -\sqrt {2 c_{1} +2 x} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & f^{\prime }-\frac {1}{f}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & f^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & f^{\prime }=\frac {1}{f} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & f^{\prime } f=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int f^{\prime } fd x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {f^{2}}{2}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} f \\ {} & {} & \left \{f=\sqrt {2 c_{1} +2 x}, f=-\sqrt {2 c_{1} +2 x}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 23
dsolve(diff(f(x),x)=f(x)^(-1),f(x), singsol=all)
\begin{align*} f \left (x \right ) &= \sqrt {c_{1} +2 x} \\ f \left (x \right ) &= -\sqrt {c_{1} +2 x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.078 (sec). Leaf size: 38
DSolve[f'[x]==f[x]^(-1),f[x],x,IncludeSingularSolutions -> True]
\begin{align*} f(x)\to -\sqrt {2} \sqrt {x+c_1} \\ f(x)\to \sqrt {2} \sqrt {x+c_1} \\ \end{align*}