1.57 problem 57

Internal problem ID [9039]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 57
Date solved : Thursday, October 17, 2024 at 01:12:38 PM
CAS classification : [_quadrature]

Solve

\begin{align*} y^{\prime }-\sqrt {{| y|}}&=0 \end{align*}

1.57.1 Solved as first order quadrature ode

Time used: 0.519 (sec)

Integrating gives

\begin{align*} \int \frac {1}{\sqrt {{| y |}}}d y &= dx\\ 2 \left (\left \{\begin {array}{cc} -\sqrt {-y} & y \le 0 \\ \sqrt {y} & 0<y \end {array}\right .\right )&= x +c_1 \end{align*}

Singular solutions are found by solving

\begin{align*} \sqrt {{| y |}}&= 0 \end{align*}

for \(y\). This is because we had to divide by this in the above step. This gives the following singular solution(s), which also have to satisfy the given ODE.

\begin{align*} y = 0 \end{align*}

Figure 32: Slope field plot
\(y^{\prime }-\sqrt {{| y|}} = 0\)

1.57.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )-\sqrt {{| y \left (x \right )|}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\sqrt {{| y \left (x \right )|}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\frac {d}{d x}y \left (x \right )}{\sqrt {{| y \left (x \right )|}}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {\frac {d}{d x}y \left (x \right )}{\sqrt {{| y \left (x \right )|}}}d x =\int 1d x +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \left \{\begin {array}{cc} -2 \sqrt {-y \left (x \right )} & y \left (x \right )\le 0 \\ 2 \sqrt {y \left (x \right )} & 0<y \left (x \right ) \end {array}\right .=x +\mathit {C1} \end {array} \]

1.57.3 Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
<- separable successful`
 

1.57.4 Maple dsolve solution

Solving time : 0.003 (sec)
Leaf size : 29

dsolve(diff(y(x),x)-abs(y(x))^(1/2) = 0, 
       y(x),singsol=all)
 

\[ x +2 \left (\left \{\begin {array}{cc} \sqrt {-y} & y\le 0 \\ -\sqrt {y} & 0<y \end {array}\right .\right )+c_{1} = 0 \]

1.57.5 Mathematica DSolve solution

Solving time : 0.16 (sec)
Leaf size : 31

DSolve[{D[y[x],x] - Sqrt[Abs[y[x]]]==0,{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {| K[1]| }}dK[1]\&\right ][x+c_1] \\ y(x)\to 0 \\ \end{align*}