Internal problem ID [2453]
Internal file name [OUTPUT/1945_Sunday_June_05_2022_02_40_20_AM_75628296/index.tex]
Book: Ordinary Differential Equations, Robert H. Martin, 1983
Section: Problem 1.1-2, page 6
Problem number: 1.1-2 (g).
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]
\[ \boxed {y^{\prime }=\frac {t}{\sqrt {t}+1}} \]
Integrating both sides gives \begin {align*} y &= \int { \frac {t}{\sqrt {t}+1}\,\mathop {\mathrm {d}t}}\\ &= \frac {2 t^{\frac {3}{2}}}{3}-t +2 \sqrt {t}-2 \ln \left (\sqrt {t}+1\right )+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {2 t^{\frac {3}{2}}}{3}-t +2 \sqrt {t}-2 \ln \left (\sqrt {t}+1\right )+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \frac {2 t^{\frac {3}{2}}}{3}-t +2 \sqrt {t}-2 \ln \left (\sqrt {t}+1\right )+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {t}{\sqrt {t}+1} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int y^{\prime }d t =\int \frac {t}{\sqrt {t}+1}d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {2 t^{\frac {3}{2}}}{3}-t +2 \sqrt {t}-2 \ln \left (\sqrt {t}+1\right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {2 t^{\frac {3}{2}}}{3}-t +2 \sqrt {t}-2 \ln \left (\sqrt {t}+1\right )+c_{1} \end {array} \]
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: 27
dsolve(diff(y(t),t)=t/(sqrt(t)+1),y(t), singsol=all)
\[ y \left (t \right ) = \frac {2 t^{\frac {3}{2}}}{3}-t +2 \sqrt {t}-2 \ln \left (\sqrt {t}+1\right )+c_{1} \]
✓ Solution by Mathematica
Time used: 0.012 (sec). Leaf size: 25
DSolve[y'[t]==1/(1+Sqrt[t]),y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to 2 \sqrt {t}-2 \log \left (\sqrt {t}+1\right )+c_1 \]