Internal problem ID [1665]
Internal file name [OUTPUT/1666_Sunday_June_05_2022_02_26_22_AM_89665322/index.tex]
Book: Differential equations and their applications, 3rd ed., M. Braun
Section: Section 1.2. Page 9
Problem number: 21.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_linear]
\[ \boxed {y^{\prime }+\frac {y}{\sqrt {t}}={\mathrm e}^{\frac {\sqrt {t}}{2}}} \]
Entering Linear first order ODE solver. In canonical form a linear first order is \begin {align*} y^{\prime } + p(t)y &= q(t) \end {align*}
Where here \begin {align*} p(t) &=\frac {1}{\sqrt {t}}\\ q(t) &={\mathrm e}^{\frac {\sqrt {t}}{2}} \end {align*}
Hence the ode is \begin {align*} y^{\prime }+\frac {y}{\sqrt {t}} = {\mathrm e}^{\frac {\sqrt {t}}{2}} \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int \frac {1}{\sqrt {t}}d t} \\ &= {\mathrm e}^{2 \sqrt {t}} \\ \end{align*} The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}t}}\left ( \mu y\right ) &= \left (\mu \right ) \left ({\mathrm e}^{\frac {\sqrt {t}}{2}}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}t}} \left ({\mathrm e}^{2 \sqrt {t}} y\right ) &= \left ({\mathrm e}^{2 \sqrt {t}}\right ) \left ({\mathrm e}^{\frac {\sqrt {t}}{2}}\right )\\ \mathrm {d} \left ({\mathrm e}^{2 \sqrt {t}} y\right ) &= {\mathrm e}^{\frac {5 \sqrt {t}}{2}}\, \mathrm {d} t \end {align*}
Integrating gives \begin {align*} {\mathrm e}^{2 \sqrt {t}} y &= \int {{\mathrm e}^{\frac {5 \sqrt {t}}{2}}\,\mathrm {d} t}\\ {\mathrm e}^{2 \sqrt {t}} y &= \frac {4 \,{\mathrm e}^{\frac {5 \sqrt {t}}{2}} \sqrt {t}}{5}-\frac {8 \,{\mathrm e}^{\frac {5 \sqrt {t}}{2}}}{25} + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu ={\mathrm e}^{2 \sqrt {t}}\) results in \begin {align*} y &= {\mathrm e}^{-2 \sqrt {t}} \left (\frac {4 \,{\mathrm e}^{\frac {5 \sqrt {t}}{2}} \sqrt {t}}{5}-\frac {8 \,{\mathrm e}^{\frac {5 \sqrt {t}}{2}}}{25}\right )+c_{1} {\mathrm e}^{-2 \sqrt {t}} \end {align*}
which simplifies to \begin {align*} y &= \frac {\left (20 \,{\mathrm e}^{\frac {5 \sqrt {t}}{2}} \sqrt {t}-8 \,{\mathrm e}^{\frac {5 \sqrt {t}}{2}}+25 c_{1} \right ) {\mathrm e}^{-2 \sqrt {t}}}{25} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (20 \,{\mathrm e}^{\frac {5 \sqrt {t}}{2}} \sqrt {t}-8 \,{\mathrm e}^{\frac {5 \sqrt {t}}{2}}+25 c_{1} \right ) {\mathrm e}^{-2 \sqrt {t}}}{25} \\ \end{align*}
Verification of solutions
\[ y = \frac {\left (20 \,{\mathrm e}^{\frac {5 \sqrt {t}}{2}} \sqrt {t}-8 \,{\mathrm e}^{\frac {5 \sqrt {t}}{2}}+25 c_{1} \right ) {\mathrm e}^{-2 \sqrt {t}}}{25} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+\frac {y}{\sqrt {t}}={\mathrm e}^{\frac {\sqrt {t}}{2}} \\ \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 {y}{\sqrt {t}}+{\mathrm e}^{\frac {\sqrt {t}}{2}} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE}\hspace {3pt} \\ {} & {} & y^{\prime }+\frac {y}{\sqrt {t}}={\mathrm e}^{\frac {\sqrt {t}}{2}} \\ \bullet & {} & \textrm {The ODE is linear; multiply by an integrating factor}\hspace {3pt} \mu \left (t \right ) \\ {} & {} & \mu \left (t \right ) \left (y^{\prime }+\frac {y}{\sqrt {t}}\right )=\mu \left (t \right ) {\mathrm e}^{\frac {\sqrt {t}}{2}} \\ \bullet & {} & \textrm {Assume the lhs of the ODE is the total derivative}\hspace {3pt} \frac {d}{d t}\left (y \mu \left (t \right )\right ) \\ {} & {} & \mu \left (t \right ) \left (y^{\prime }+\frac {y}{\sqrt {t}}\right )=y^{\prime } \mu \left (t \right )+y \mu ^{\prime }\left (t \right ) \\ \bullet & {} & \textrm {Isolate}\hspace {3pt} \mu ^{\prime }\left (t \right ) \\ {} & {} & \mu ^{\prime }\left (t \right )=\frac {\mu \left (t \right )}{\sqrt {t}} \\ \bullet & {} & \textrm {Solve to find the integrating factor}\hspace {3pt} \\ {} & {} & \mu \left (t \right )={\mathrm e}^{2 \sqrt {t}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \left (\frac {d}{d t}\left (y \mu \left (t \right )\right )\right )d t =\int \mu \left (t \right ) {\mathrm e}^{\frac {\sqrt {t}}{2}}d t +c_{1} \\ \bullet & {} & \textrm {Evaluate the integral on the lhs}\hspace {3pt} \\ {} & {} & y \mu \left (t \right )=\int \mu \left (t \right ) {\mathrm e}^{\frac {\sqrt {t}}{2}}d t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\int \mu \left (t \right ) {\mathrm e}^{\frac {\sqrt {t}}{2}}d t +c_{1}}{\mu \left (t \right )} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \mu \left (t \right )={\mathrm e}^{2 \sqrt {t}} \\ {} & {} & y=\frac {\int {\mathrm e}^{\frac {\sqrt {t}}{2}} {\mathrm e}^{2 \sqrt {t}}d t +c_{1}}{{\mathrm e}^{2 \sqrt {t}}} \\ \bullet & {} & \textrm {Evaluate the integrals on the rhs}\hspace {3pt} \\ {} & {} & y=\frac {\frac {4 \,{\mathrm e}^{\frac {5 \sqrt {t}}{2}} \sqrt {t}}{5}-\frac {8 \,{\mathrm e}^{\frac {5 \sqrt {t}}{2}}}{25}+c_{1}}{{\mathrm e}^{2 \sqrt {t}}} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & y=\frac {\left (20 \,{\mathrm e}^{\frac {5 \sqrt {t}}{2}} \sqrt {t}-8 \,{\mathrm e}^{\frac {5 \sqrt {t}}{2}}+25 c_{1} \right ) {\mathrm e}^{-2 \sqrt {t}}}{25} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 35
dsolve(diff(y(t),t)+1/sqrt(t)*y(t)=exp(sqrt(t)/2),y(t), singsol=all)
\[ y \left (t \right ) = \frac {\left (20 \,{\mathrm e}^{\frac {5 \sqrt {t}}{2}} \sqrt {t}-8 \,{\mathrm e}^{\frac {5 \sqrt {t}}{2}}+25 c_{1} \right ) {\mathrm e}^{-2 \sqrt {t}}}{25} \]
✓ Solution by Mathematica
Time used: 0.093 (sec). Leaf size: 42
DSolve[y'[t]+1/Sqrt[t]*y[t]==Exp[Sqrt[t]/2],y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \frac {4}{25} e^{\frac {\sqrt {t}}{2}} \left (5 \sqrt {t}-2\right )+c_1 e^{-2 \sqrt {t}} \]