1.49 problem 49

Internal problem ID [8013]
Book : First order enumerated odes
Section : section 1
Problem number : 49
Date solved : Monday, October 21, 2024 at 04:40:54 PM
CAS classification : [_quadrature]

Solve

\begin{align*} {y^{\prime }}^{2}&=x \end{align*}

Solving for the derivative gives these ODE’s to solve

\begin{align*} \tag{1} y^{\prime }&=\sqrt {x} \\ \tag{2} y^{\prime }&=-\sqrt {x} \\ \end{align*}

Now each of the above is solved separately.

Solving Eq. (1)

Since the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\).

\begin{align*} \int {dy} &= \int {\sqrt {x}\, dx}\\ y &= \frac {2 x^{{3}/{2}}}{3} + c_1 \end{align*}

Solving Eq. (2)

Since the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\).

\begin{align*} \int {dy} &= \int {-\sqrt {x}\, dx}\\ y &= -\frac {2 x^{{3}/{2}}}{3} + c_2 \end{align*}

1.49.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{2}=x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\sqrt {x}, y^{\prime }=-\sqrt {x}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\sqrt {x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \sqrt {x}d x +\textit {\_C1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {2 x^{{3}/{2}}}{3}+\textit {\_C1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {2 x^{{3}/{2}}}{3}+\textit {\_C1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\sqrt {x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int -\sqrt {x}d x +\textit {\_C1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\frac {2 x^{{3}/{2}}}{3}+\textit {\_C1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {2 x^{{3}/{2}}}{3}+\textit {\_C1} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=-\frac {2 x^{{3}/{2}}}{3}+\mathit {C1} , y=\frac {2 x^{{3}/{2}}}{3}+\mathit {C1} \right \} \end {array} \]

1.49.2 Maple trace

Methods for first order ODEs:
 

1.49.3 Maple dsolve solution

Solving time : 0.022 (sec)
Leaf size : 21

dsolve(diff(y(x),x)^2 = x, 
       y(x),singsol=all)
 

\begin{align*} y &= \frac {2 x^{{3}/{2}}}{3}+c_1 \\ y &= -\frac {2 x^{{3}/{2}}}{3}+c_1 \\ \end{align*}

1.49.4 Mathematica DSolve solution

Solving time : 0.004 (sec)
Leaf size : 33

DSolve[{(D[y[x],x])^2==x,{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {2 x^{3/2}}{3}+c_1 \\ y(x)\to \frac {2 x^{3/2}}{3}+c_1 \\ \end{align*}