Internal problem ID [11361]
Internal file name [OUTPUT/10344_Wednesday_May_17_2023_07_49_34_PM_52810511/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: 3.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "kovacic", "exact linear second order ode", "second_order_integrable_as_is", "second_order_ode_missing_y", "second_order_ode_quadrature", "second_order_linear_constant_coeff"
Maple gives the following as the ode type
[[_2nd_order, _quadrature]]
\[ \boxed {x^{\prime \prime }=-3 \sqrt {t}} \] With initial conditions \begin {align*} [x \left (1\right ) = 4, x^{\prime }\left (1\right ) = 2] \end {align*}
This is a linear ODE. In canonical form it is written as \begin {align*} x^{\prime \prime } + p(t)x^{\prime } + q(t) x &= F \end {align*}
Where here \begin {align*} p(t) &=0\\ q(t) &=0\\ F &=-3 \sqrt {t} \end {align*}
Hence the ode is \begin {align*} x^{\prime \prime } = -3 \sqrt {t} \end {align*}
The domain of \(p(t)=0\) is \[
\{-\infty
Integrating once gives \[ x^{\prime }= -2 t^{\frac {3}{2}} + c_{1} \] Integrating again gives \[ x= -\frac {4 t^{\frac {5}{2}}}{5} + c_{1} x + c_{2} \] Initial conditions are used to solve for the
constants of integration.
Looking at the above solution \begin {align*} x = -\frac {4 t^{\frac {5}{2}}}{5}+c_{1} t +c_{2} \tag {1} \end {align*}
Initial conditions are now substituted in the above solution. This will generate the required
equations to solve for the integration constants. substituting \(x = 4\) and \(t = 1\) in the above gives
\begin {align*} 4 = -\frac {4}{5}+c_{1} +c_{2}\tag {1A} \end {align*}
Taking derivative of the solution gives \begin {align*} x^{\prime } = -2 t^{\frac {3}{2}}+c_{1} \end {align*}
substituting \(x^{\prime } = 2\) and \(t = 1\) in the above gives \begin {align*} 2 = -2+c_{1}\tag {2A} \end {align*}
Equations {1A,2A} are now solved for \(\{c_{1}, c_{2}\}\). Solving for the constants gives \begin {align*} c_{1}&=4\\ c_{2}&={\frac {4}{5}} \end {align*}
Substituting these values back in above solution results in \begin {align*} x = -\frac {4 t^{\frac {5}{2}}}{5}+4 t +\frac {4}{5} \end {align*}
Summary
The solution(s) found are the following \begin{align*}
\tag{1} x &= -\frac {4 t^{\frac {5}{2}}}{5}+4 t +\frac {4}{5} \\
\end{align*} Verification of solutions
\[
x = -\frac {4 t^{\frac {5}{2}}}{5}+4 t +\frac {4}{5}
\] Verified OK. This is second order non-homogeneous ODE. In standard form the ODE is \[ A x''(t) + B x'(t) + C x(t) = f(t) \] Where \(A=1, B=0, C=0, f(t)=-3 \sqrt {t}\).
Let the solution be \[ x = x_h + x_p \] Where \(x_h\) is the solution to the homogeneous ODE \( A x''(t) + B x'(t) + C x(t) = 0\), and \(x_p\) is a
particular solution to the non-homogeneous ODE \(A x''(t) + B x'(t) + C x(t) = f(t)\). \(x_h\) is the solution to \[ x^{\prime \prime } = 0 \] This is second
order with constant coefficients homogeneous ODE. In standard form the ODE is \[ A x''(t) + B x'(t) + C x(t) = 0 \]
Where in the above \(A=1, B=0, C=0\). Let the solution be \(x=e^{\lambda t}\). Substituting this into the ODE gives \[ \lambda ^{2} {\mathrm e}^{\lambda t} = 0 \tag {1} \]
Since exponential function is never zero, then dividing Eq(2) throughout by \(e^{\lambda t}\) gives \[ \lambda ^{2} = 0 \tag {2} \]
Equation (2) is the characteristic equation of the ODE. Its roots determine the
general solution form.Using the quadratic formula \[ \lambda _{1,2} = \frac {-B}{2 A} \pm \frac {1}{2 A} \sqrt {B^2 - 4 A C} \] Substituting \(A=1, B=0, C=0\) into the above gives
\begin {align*} \lambda _{1,2} &= \frac {0}{(2) \left (1\right )} \pm \frac {1}{(2) \left (1\right )} \sqrt {\left (0\right )^2 - (4) \left (1\right )\left (0\right )}\\ &= 0 \end {align*}
Hence this is the case of a double root \(\lambda _{1,2} = 0\). Therefore the solution is \[ x= c_{1} 1 + c_{2} t \tag {1} \] Therefore the
homogeneous solution \(x_h\) is \[
x_h = c_{2} t +c_{1}
\] The particular solution \(x_p\) can be found using either the method of
undetermined coefficients, or the method of variation of parameters. The method of variation
of parameters will be used as it is more general and can be used when the coefficients of the
ODE depend on \(t\) as well. Let \begin{equation}
\tag{1} x_p(t) = u_1 x_1 + u_2 x_2
\end{equation} Where \(u_1,u_2\) to be determined, and \(x_1,x_2\) are the two basis solutions (the
two linearly independent solutions of the homogeneous ODE) found earlier when solving the
homogeneous ODE as \begin{align*}
x_1 &= 1 \\
x_2 &= t \\
\end{align*} In the Variation of parameters \(u_1,u_2\) are found using \begin{align*}
\tag{2} u_1 &= -\int \frac {x_2 f(t)}{a W(t)} \\
\tag{3} u_2 &= \int \frac {x_1 f(t)}{a W(t)} \\
\end{align*} Where \(W(t)\) is the
Wronskian and \(a\) is the coefficient in front of \(x''\) in the given ODE. The Wronskian is given
by \(W= \begin {vmatrix} x_1 & x_{2} \\ x_{1}^{\prime } & x_{2}^{\prime } \end {vmatrix} \). Hence \[ W = \begin {vmatrix} 1 & t \\ \frac {d}{dt}\left (1\right ) & \frac {d}{dt}\left (t\right ) \end {vmatrix} \] Which gives \[ W = \begin {vmatrix} 1 & t \\ 0 & 1 \end {vmatrix} \] Therefore \[
W = \left (1\right )\left (1\right ) - \left (t\right )\left (0\right )
\] Which simplifies to \[
W = 1
\] Which simplifies to \[
W = 1
\]
Therefore Eq. (2) becomes \[
u_1 = -\int \frac {-3 t^{\frac {3}{2}}}{1}\,dt
\] Which simplifies to \[
u_1 = - \int -3 t^{\frac {3}{2}}d t
\] Hence \[
u_1 = \frac {6 t^{\frac {5}{2}}}{5}
\] And Eq. (3) becomes \[
u_2 = \int \frac {-3 \sqrt {t}}{1}\,dt
\] Which
simplifies to \[
u_2 = \int -3 \sqrt {t}d t
\] Hence \[
u_2 = -2 t^{\frac {3}{2}}
\] Therefore the particular solution, from equation (1) is \[
x_p(t) = -\frac {4 t^{\frac {5}{2}}}{5}
\] Therefore
the general solution is \begin{align*}
x &= x_h + x_p \\
&= \left (c_{2} t +c_{1}\right ) + \left (-\frac {4 t^{\frac {5}{2}}}{5}\right ) \\
\end{align*} Initial conditions are used to solve for the constants of
integration.
Looking at the above solution \begin {align*} x = c_{2} t +c_{1} -\frac {4 t^{\frac {5}{2}}}{5} \tag {1} \end {align*}
Initial conditions are now substituted in the above solution. This will generate the required
equations to solve for the integration constants. substituting \(x = 4\) and \(t = 1\) in the above gives
\begin {align*} 4 = -\frac {4}{5}+c_{1} +c_{2}\tag {1A} \end {align*}
Taking derivative of the solution gives \begin {align*} x^{\prime } = c_{2} -2 t^{\frac {3}{2}} \end {align*}
substituting \(x^{\prime } = 2\) and \(t = 1\) in the above gives \begin {align*} 2 = c_{2} -2\tag {2A} \end {align*}
Equations {1A,2A} are now solved for \(\{c_{1}, c_{2}\}\). Solving for the constants gives \begin {align*} c_{1}&={\frac {4}{5}}\\ c_{2}&=4 \end {align*}
Substituting these values back in above solution results in \begin {align*} x = -\frac {4 t^{\frac {5}{2}}}{5}+4 t +\frac {4}{5} \end {align*}
Summary
The solution(s) found are the following \begin{align*}
\tag{1} x &= -\frac {4 t^{\frac {5}{2}}}{5}+4 t +\frac {4}{5} \\
\end{align*} Verification of solutions
\[
x = -\frac {4 t^{\frac {5}{2}}}{5}+4 t +\frac {4}{5}
\] Verified OK. Integrating both sides of the ODE w.r.t \(t\) gives \begin {align*} \int x^{\prime \prime }d t &= \int -3 \sqrt {t}d t\\ x^{\prime } = -2 t^{\frac {3}{2}} + c_{1} \end {align*}
Which is now solved for \(x\). Integrating both sides gives \begin {align*} x &= \int { -2 t^{\frac {3}{2}}+c_{1}\,\mathop {\mathrm {d}t}}\\ &= -\frac {4 t^{\frac {5}{2}}}{5}+c_{1} t +c_{2} \end {align*}
Initial conditions are used to solve for the constants of integration.
Looking at the above solution \begin {align*} x = -\frac {4 t^{\frac {5}{2}}}{5}+c_{1} t +c_{2} \tag {1} \end {align*}
Initial conditions are now substituted in the above solution. This will generate the required
equations to solve for the integration constants. substituting \(x = 4\) and \(t = 1\) in the above gives
\begin {align*} 4 = -\frac {4}{5}+c_{1} +c_{2}\tag {1A} \end {align*}
Taking derivative of the solution gives \begin {align*} x^{\prime } = -2 t^{\frac {3}{2}}+c_{1} \end {align*}
substituting \(x^{\prime } = 2\) and \(t = 1\) in the above gives \begin {align*} 2 = -2+c_{1}\tag {2A} \end {align*}
Equations {1A,2A} are now solved for \(\{c_{1}, c_{2}\}\). Solving for the constants gives \begin {align*} c_{1}&=4\\ c_{2}&={\frac {4}{5}} \end {align*}
Substituting these values back in above solution results in \begin {align*} x = -\frac {4 t^{\frac {5}{2}}}{5}+4 t +\frac {4}{5} \end {align*}
Summary
The solution(s) found are the following \begin{align*}
\tag{1} x &= -\frac {4 t^{\frac {5}{2}}}{5}+4 t +\frac {4}{5} \\
\end{align*} Verification of solutions
\[
x = -\frac {4 t^{\frac {5}{2}}}{5}+4 t +\frac {4}{5}
\] Verified OK. This is second order ode with missing dependent variable \(x\). Let \begin {align*} p(t) &= x^{\prime } \end {align*}
Then \begin {align*} p'(t) &= x^{\prime \prime } \end {align*}
Hence the ode becomes \begin {align*} p^{\prime }\left (t \right )+3 \sqrt {t} = 0 \end {align*}
Which is now solve for \(p(t)\) as first order ode. Integrating both sides gives \begin {align*} p \left (t \right ) &= \int { -3 \sqrt {t}\,\mathop {\mathrm {d}t}}\\ &= -2 t^{\frac {3}{2}}+c_{1} \end {align*}
Initial conditions are used to solve for \(c_{1}\). Substituting \(t=1\) and \(p=2\) in the above solution gives an
equation to solve for the constant of integration. \begin {align*} 2 = -2+c_{1} \end {align*}
The solutions are \begin {align*} c_{1} = 4 \end {align*}
Trying the constant \begin {align*} c_{1} = 4 \end {align*}
Substituting this in the general solution gives \begin {align*} p \left (t \right )&=-2 t^{\frac {3}{2}}+4 \end {align*}
The constant \(c_{1} = 4\) gives valid solution.
Since \(p=x^{\prime }\) then the new first order ode to solve is \begin {align*} x^{\prime } = -2 t^{\frac {3}{2}}+4 \end {align*}
Integrating both sides gives \begin {align*} x &= \int { -2 t^{\frac {3}{2}}+4\,\mathop {\mathrm {d}t}}\\ &= -\frac {4 t^{\frac {5}{2}}}{5}+4 t +c_{2} \end {align*}
Initial conditions are used to solve for \(c_{2}\). Substituting \(t=1\) and \(x=4\) in the above solution gives an
equation to solve for the constant of integration. \begin {align*} 4 = \frac {16}{5}+c_{2} \end {align*}
The solutions are \begin {align*} c_{2} = {\frac {4}{5}} \end {align*}
Trying the constant \begin {align*} c_{2} = {\frac {4}{5}} \end {align*}
Substituting this in the general solution gives \begin {align*} x&=-\frac {4 t^{\frac {5}{2}}}{5}+4 t +\frac {4}{5} \end {align*}
The constant \(c_{2} = {\frac {4}{5}}\) gives valid solution.
Initial conditions are used to solve for the constants of integration.
Summary
The solution(s) found are the following \begin{align*}
\tag{1} x &= -\frac {4 t^{\frac {5}{2}}}{5}+4 t +\frac {4}{5} \\
\end{align*} Verification of solutions
\[
x = -\frac {4 t^{\frac {5}{2}}}{5}+4 t +\frac {4}{5}
\] Verified OK. Writing the ode as \begin {align*} x^{\prime \prime } &= 0 \tag {1} \\ A x^{\prime \prime } + B x^{\prime } + C x &= 0 \tag {2} \end {align*}
Comparing (1) and (2) shows that \begin {align*} A &= 1 \\ B &= 0\tag {3} \\ C &= 0 \end {align*}
Applying the Liouville transformation on the dependent variable gives \begin {align*} z(t) &= x e^{\int \frac {B}{2 A} \,dt} \end {align*}
Then (2) becomes \begin {align*} z''(t) = r z(t)\tag {4} \end {align*}
Where \(r\) is given by \begin {align*} r &= \frac {s}{t}\tag {5} \\ &= \frac {2 A B' - 2 B A' + B^2 - 4 A C}{4 A^2} \end {align*}
Substituting the values of \(A,B,C\) from (3) in the above and simplifying gives \begin {align*} r &= \frac {0}{1}\tag {6} \end {align*}
Comparing the above to (5) shows that \begin {align*} s &= 0\\ t &= 1 \end {align*}
Therefore eq. (4) becomes \begin {align*} z''(t) &= 0 \tag {7} \end {align*}
Equation (7) is now solved. After finding \(z(t)\) then \(x\) is found using the inverse transformation
\begin {align*} x &= z \left (t \right ) e^{-\int \frac {B}{2 A} \,dt} \end {align*}
The first step is to determine the case of Kovacic algorithm this ode belongs to. There are 3
cases depending on the order of poles of \(r\) and the order of \(r\) at \(\infty \). The following table
summarizes these cases. Case Allowed pole order for \(r\) Allowed value for \(\mathcal {O}(\infty )\) 1 \(\left \{ 0,1,2,4,6,8,\cdots \right \} \) \(\left \{ \cdots ,-6,-4,-2,0,2,3,4,5,6,\cdots \right \} \) 2
Need to have at least one pole
that is either order \(2\) or odd order
greater than \(2\). Any other pole order
is allowed as long as the above
condition is satisfied. Hence the
following set of pole orders are all
allowed. \(\{1,2\}\),\(\{1,3\}\),\(\{2\}\),\(\{3\}\),\(\{3,4\}\),\(\{1,2,5\}\). no condition 3 \(\left \{ 1,2\right \} \) \(\left \{ 2,3,4,5,6,7,\cdots \right \} \) The order of \(r\) at \(\infty \) is the degree of \(t\) minus the degree of \(s\). Therefore \begin {align*} O\left (\infty \right ) &= \text {deg}(t) - \text {deg}(s) \\ &= 0 - -\infty \\ &= \infty \end {align*}
There are no poles in \(r\). Therefore the set of poles \(\Gamma \) is empty. Since there is no odd order pole
larger than \(2\) and the order at \(\infty \) is \(infinity\) then the necessary conditions for case one are met.
Therefore \begin {align*} L &= [1] \end {align*}
Since \(r = 0\) is not a function of \(t\), then there is no need run Kovacic algorithm to obtain a solution
for transformed ode \(z''=r z\) as one solution is \[ z_1(t) = 1 \] Using the above, the solution for the original
ode can now be found. The first solution to the original ode in \(x\) is found from
\[
x_1 = z_1 e^{ \int -\frac {1}{2} \frac {B}{A} \,dt}
\]
Since \(B=0\) then the above reduces to \begin{align*}
x_1 &= z_1 \\
&= 1 \\
\end{align*} Which simplifies to \[
x_1 = 1
\] The second solution \(x_2\) to the original
ode is found using reduction of order \[ x_2 = x_1 \int \frac { e^{\int -\frac {B}{A} \,dt}}{x_1^2} \,dt \] Since \(B=0\) then the above becomes \begin{align*}
x_2 &= x_1 \int \frac {1}{x_1^2} \,dt \\
&= 1\int \frac {1}{1} \,dt \\
&= 1\left (t\right ) \\
\end{align*} Therefore the solution
is
\begin{align*}
x &= c_{1} x_1 + c_{2} x_2 \\
&= c_{1} \left (1\right ) + c_{2} \left (1\left (t\right )\right ) \\
\end{align*} This is second order nonhomogeneous ODE. Let the solution be \[
x = x_h + x_p
\] Where \(x_h\) is the solution to
the homogeneous ODE \( A x''(t) + B x'(t) + C x(t) = 0\), and \(x_p\) is a particular solution to the nonhomogeneous ODE \(A x''(t) + B x'(t) + C x(t) = f(t)\). \(x_h\) is the
solution to \[
x^{\prime \prime } = 0
\] The homogeneous solution is found using the Kovacic algorithm which results in \[
x_h = c_{2} t +c_{1}
\]
The particular solution \(x_p\) can be found using either the method of undetermined coefficients,
or the method of variation of parameters. The method of variation of parameters will be
used as it is more general and can be used when the coefficients of the ODE depend on \(t\) as
well. Let \begin{equation}
\tag{1} x_p(t) = u_1 x_1 + u_2 x_2
\end{equation} Where \(u_1,u_2\) to be determined, and \(x_1,x_2\) are the two basis solutions (the two linearly
independent solutions of the homogeneous ODE) found earlier when solving the
homogeneous ODE as \begin{align*}
x_1 &= 1 \\
x_2 &= t \\
\end{align*} In the Variation of parameters \(u_1,u_2\) are found using \begin{align*}
\tag{2} u_1 &= -\int \frac {x_2 f(t)}{a W(t)} \\
\tag{3} u_2 &= \int \frac {x_1 f(t)}{a W(t)} \\
\end{align*} Where \(W(t)\) is the
Wronskian and \(a\) is the coefficient in front of \(x''\) in the given ODE. The Wronskian is given
by \(W= \begin {vmatrix} x_1 & x_{2} \\ x_{1}^{\prime } & x_{2}^{\prime } \end {vmatrix} \). Hence \[ W = \begin {vmatrix} 1 & t \\ \frac {d}{dt}\left (1\right ) & \frac {d}{dt}\left (t\right ) \end {vmatrix} \] Which gives \[ W = \begin {vmatrix} 1 & t \\ 0 & 1 \end {vmatrix} \] Therefore \[
W = \left (1\right )\left (1\right ) - \left (t\right )\left (0\right )
\] Which simplifies to \[
W = 1
\] Which simplifies to \[
W = 1
\]
Therefore Eq. (2) becomes \[
u_1 = -\int \frac {-3 t^{\frac {3}{2}}}{1}\,dt
\] Which simplifies to \[
u_1 = - \int -3 t^{\frac {3}{2}}d t
\] Hence \[
u_1 = \frac {6 t^{\frac {5}{2}}}{5}
\] And Eq. (3) becomes \[
u_2 = \int \frac {-3 \sqrt {t}}{1}\,dt
\] Which
simplifies to \[
u_2 = \int -3 \sqrt {t}d t
\] Hence \[
u_2 = -2 t^{\frac {3}{2}}
\] Therefore the particular solution, from equation (1) is \[
x_p(t) = -\frac {4 t^{\frac {5}{2}}}{5}
\] Therefore
the general solution is \begin{align*}
x &= x_h + x_p \\
&= \left (c_{2} t +c_{1}\right ) + \left (-\frac {4 t^{\frac {5}{2}}}{5}\right ) \\
\end{align*} Initial conditions are used to solve for the constants of
integration.
Looking at the above solution \begin {align*} x = c_{2} t +c_{1} -\frac {4 t^{\frac {5}{2}}}{5} \tag {1} \end {align*}
Initial conditions are now substituted in the above solution. This will generate the required
equations to solve for the integration constants. substituting \(x = 4\) and \(t = 1\) in the above gives
\begin {align*} 4 = -\frac {4}{5}+c_{1} +c_{2}\tag {1A} \end {align*}
Taking derivative of the solution gives \begin {align*} x^{\prime } = c_{2} -2 t^{\frac {3}{2}} \end {align*}
substituting \(x^{\prime } = 2\) and \(t = 1\) in the above gives \begin {align*} 2 = c_{2} -2\tag {2A} \end {align*}
Equations {1A,2A} are now solved for \(\{c_{1}, c_{2}\}\). Solving for the constants gives \begin {align*} c_{1}&={\frac {4}{5}}\\ c_{2}&=4 \end {align*}
Substituting these values back in above solution results in \begin {align*} x = -\frac {4 t^{\frac {5}{2}}}{5}+4 t +\frac {4}{5} \end {align*}
Summary
The solution(s) found are the following \begin{align*}
\tag{1} x &= -\frac {4 t^{\frac {5}{2}}}{5}+4 t +\frac {4}{5} \\
\end{align*} Verification of solutions
\[
x = -\frac {4 t^{\frac {5}{2}}}{5}+4 t +\frac {4}{5}
\] Verified OK. An ode of the form \begin {align*} p \left (t \right ) x^{\prime \prime }+q \left (t \right ) x^{\prime }+r \left (t \right ) x&=s \left (t \right ) \end {align*}
is exact if \begin {align*} p''(t) - q'(t) + r(t) &= 0 \tag {1} \end {align*}
For the given ode we have \begin {align*} p(x) &= 1\\ q(x) &= 0\\ r(x) &= 0\\ s(x) &= -3 \sqrt {t} \end {align*}
Hence \begin {align*} p''(x) &= 0\\ q'(x) &= 0 \end {align*}
Therefore (1) becomes \begin {align*} 0- \left (0\right ) + \left (0\right )&=0 \end {align*}
Hence the ode is exact. Since we now know the ode is exact, it can be written as
\begin {align*} \left (p \left (t \right ) x^{\prime }+\left (q \left (t \right )-p^{\prime }\left (t \right )\right ) x\right )' &= s(x) \end {align*}
Integrating gives \begin {align*} p \left (t \right ) x^{\prime }+\left (q \left (t \right )-p^{\prime }\left (t \right )\right ) x&=\int {s \left (t \right )\, dt} \end {align*}
Substituting the above values for \(p,q,r,s\) gives \begin {align*} x^{\prime }&=\int {-3 \sqrt {t}\, dt} \end {align*}
We now have a first order ode to solve which is \begin {align*} x^{\prime } = -2 t^{\frac {3}{2}}+c_{1} \end {align*}
Integrating both sides gives \begin {align*} x &= \int { -2 t^{\frac {3}{2}}+c_{1}\,\mathop {\mathrm {d}t}}\\ &= -\frac {4 t^{\frac {5}{2}}}{5}+c_{1} t +c_{2} \end {align*}
Initial conditions are used to solve for the constants of integration.
Looking at the above solution \begin {align*} x = -\frac {4 t^{\frac {5}{2}}}{5}+c_{1} t +c_{2} \tag {1} \end {align*}
Initial conditions are now substituted in the above solution. This will generate the required
equations to solve for the integration constants. substituting \(x = 4\) and \(t = 1\) in the above gives
\begin {align*} 4 = -\frac {4}{5}+c_{1} +c_{2}\tag {1A} \end {align*}
Taking derivative of the solution gives \begin {align*} x^{\prime } = -2 t^{\frac {3}{2}}+c_{1} \end {align*}
substituting \(x^{\prime } = 2\) and \(t = 1\) in the above gives \begin {align*} 2 = -2+c_{1}\tag {2A} \end {align*}
Equations {1A,2A} are now solved for \(\{c_{1}, c_{2}\}\). Solving for the constants gives \begin {align*} c_{1}&=4\\ c_{2}&={\frac {4}{5}} \end {align*}
Substituting these values back in above solution results in \begin {align*} x = -\frac {4 t^{\frac {5}{2}}}{5}+4 t +\frac {4}{5} \end {align*}
Summary
The solution(s) found are the following \begin{align*}
\tag{1} x &= -\frac {4 t^{\frac {5}{2}}}{5}+4 t +\frac {4}{5} \\
\end{align*} Verification of solutions
\[
x = -\frac {4 t^{\frac {5}{2}}}{5}+4 t +\frac {4}{5}
\] Verified OK. \[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [x^{\prime \prime }=-3 \sqrt {t}, x \left (1\right )=4, x^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}1\right \}}}}=2\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & x^{\prime \prime } \\ \bullet & {} & \textrm {Characteristic polynomial of homogeneous ODE}\hspace {3pt} \\ {} & {} & r^{2}=0 \\ \bullet & {} & \textrm {Use quadratic formula to solve for}\hspace {3pt} r \\ {} & {} & r =\frac {0\pm \left (\sqrt {0}\right )}{2} \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =0 \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & x_{1}\left (t \right )=1 \\ \bullet & {} & \textrm {Repeated root, multiply}\hspace {3pt} x_{1}\left (t \right )\hspace {3pt}\textrm {by}\hspace {3pt} t \hspace {3pt}\textrm {to ensure linear independence}\hspace {3pt} \\ {} & {} & x_{2}\left (t \right )=t \\ \bullet & {} & \textrm {General solution of the ODE}\hspace {3pt} \\ {} & {} & x=c_{1} x_{1}\left (t \right )+c_{2} x_{2}\left (t \right )+x_{p}\left (t \right ) \\ \bullet & {} & \textrm {Substitute in solutions of the homogeneous ODE}\hspace {3pt} \\ {} & {} & x=c_{1} +c_{2} t +x_{p}\left (t \right ) \\ \square & {} & \textrm {Find a particular solution}\hspace {3pt} x_{p}\left (t \right )\hspace {3pt}\textrm {of the ODE}\hspace {3pt} \\ {} & \circ & \textrm {Use variation of parameters to find}\hspace {3pt} x_{p}\hspace {3pt}\textrm {here}\hspace {3pt} f \left (t \right )\hspace {3pt}\textrm {is the forcing function}\hspace {3pt} \\ {} & {} & \left [x_{p}\left (t \right )=-x_{1}\left (t \right ) \left (\int \frac {x_{2}\left (t \right ) f \left (t \right )}{W \left (x_{1}\left (t \right ), x_{2}\left (t \right )\right )}d t \right )+x_{2}\left (t \right ) \left (\int \frac {x_{1}\left (t \right ) f \left (t \right )}{W \left (x_{1}\left (t \right ), x_{2}\left (t \right )\right )}d t \right ), f \left (t \right )=-3 \sqrt {t}\right ] \\ {} & \circ & \textrm {Wronskian of solutions of the homogeneous equation}\hspace {3pt} \\ {} & {} & W \left (x_{1}\left (t \right ), x_{2}\left (t \right )\right )=\left [\begin {array}{cc} 1 & t \\ 0 & 1 \end {array}\right ] \\ {} & \circ & \textrm {Compute Wronskian}\hspace {3pt} \\ {} & {} & W \left (x_{1}\left (t \right ), x_{2}\left (t \right )\right )=1 \\ {} & \circ & \textrm {Substitute functions into equation for}\hspace {3pt} x_{p}\left (t \right ) \\ {} & {} & x_{p}\left (t \right )=3 \left (\int t^{\frac {3}{2}}d t \right )-3 t \left (\int \sqrt {t}d t \right ) \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & x_{p}\left (t \right )=-\frac {4 t^{\frac {5}{2}}}{5} \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & x=c_{2} t +c_{1} -\frac {4 t^{\frac {5}{2}}}{5} \\ \square & {} & \textrm {Check validity of solution}\hspace {3pt} x=c_{2} t +c_{1} -\frac {4 t^{\frac {5}{2}}}{5} \\ {} & \circ & \textrm {Use initial condition}\hspace {3pt} x \left (1\right )=4 \\ {} & {} & 4=-\frac {4}{5}+c_{1} +c_{2} \\ {} & \circ & \textrm {Compute derivative of the solution}\hspace {3pt} \\ {} & {} & x^{\prime }=c_{2} -2 t^{\frac {3}{2}} \\ {} & \circ & \textrm {Use the initial condition}\hspace {3pt} x^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}1\right \}}}}=2 \\ {} & {} & 2=c_{2} -2 \\ {} & \circ & \textrm {Solve for}\hspace {3pt} c_{1} \hspace {3pt}\textrm {and}\hspace {3pt} c_{2} \\ {} & {} & \left \{c_{1} =\frac {4}{5}, c_{2} =4\right \} \\ {} & \circ & \textrm {Substitute constant values into general solution and simplify}\hspace {3pt} \\ {} & {} & x=-\frac {4 t^{\frac {5}{2}}}{5}+4 t +\frac {4}{5} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & x=-\frac {4 t^{\frac {5}{2}}}{5}+4 t +\frac {4}{5} \end {array} \]
Maple trace
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 14
\[
x \left (t \right ) = -\frac {4 t^{\frac {5}{2}}}{5}+4 t +\frac {4}{5}
\]
✓ Solution by Mathematica
Time used: 0.039 (sec). Leaf size: 19
\[
x(t)\to -\frac {4}{5} \left (t^{5/2}-5 t-1\right )
\]
3.3.2 Solving as second order ode quadrature ode
3.3.3 Solving as second order linear constant coeff ode
3.3.4 Solving as second order integrable as is ode
3.3.5 Solving as second order ode missing y ode
3.3.6 Solving using Kovacic algorithm
3.3.7 Solving as exact linear second order ode ode
3.3.8 Maple step by step solution
`Methods for second order ODEs:
--- Trying classification methods ---
trying a quadrature
<- quadrature successful`
dsolve([diff(x(t),t$2)=-3*sqrt(t),x(1) = 4, D(x)(1) = 2],x(t), singsol=all)
DSolve[{x''[t]==-3*Sqrt[t],{x[1]==4,x'[1]==2}},x[t],t,IncludeSingularSolutions -> True]