Internal problem ID [11364]
Internal file name [OUTPUT/10347_Wednesday_May_17_2023_07_49_39_PM_55711235/index.tex]
Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag,
NY. 2015.
Section: Chapter 1, First order differential equations. Section 1.2 Antiderivatives. Exercises page
19
Problem number: 4(c).
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]
\[ \boxed {x^{\prime } \sqrt {t}=\cos \left (\sqrt {t}\right )} \]
Integrating both sides gives \begin {align*} x &= \int { \frac {\cos \left (\sqrt {t}\right )}{\sqrt {t}}\,\mathop {\mathrm {d}t}}\\ &= 2 \sin \left (\sqrt {t}\right )+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} x &= 2 \sin \left (\sqrt {t}\right )+c_{1} \\ \end{align*}
Verification of solutions
\[ x = 2 \sin \left (\sqrt {t}\right )+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{\prime } \sqrt {t}=\cos \left (\sqrt {t}\right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & x^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & x^{\prime }=\frac {\cos \left (\sqrt {t}\right )}{\sqrt {t}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int x^{\prime }d t =\int \frac {\cos \left (\sqrt {t}\right )}{\sqrt {t}}d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & x=2 \sin \left (\sqrt {t}\right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} x \\ {} & {} & x=2 \sin \left (\sqrt {t}\right )+c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 12
dsolve(sqrt(t)*diff(x(t),t)=cos(sqrt(t)),x(t), singsol=all)
\[ x \left (t \right ) = 2 \sin \left (\sqrt {t}\right )+c_{1} \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 16
DSolve[Sqrt[t]*x'[t]==Cos[Sqrt[t]],x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to 2 \sin \left (\sqrt {t}\right )+c_1 \]