1.45 problem 45

Internal problem ID [8009]
Book : First order enumerated odes
Section : section 1
Problem number : 45
Date solved : Monday, October 21, 2024 at 04:40:51 PM
CAS classification : [_quadrature]

Solve

\begin{align*} x \sin \left (x \right ) {y^{\prime }}^{2}&=0 \end{align*}

Solving for the derivative gives these ODE’s to solve

\begin{align*} \tag{1} y^{\prime }&=0 \\ \tag{2} y^{\prime }&=0 \\ \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 {0\, dx} + c_1 \\ y &= 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 {0\, dx} + c_2 \\ y &= c_2 \end{align*}

1.45.1 Maple step by step solution

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

1.45.2 Maple trace

Methods for first order ODEs:
 

1.45.3 Maple dsolve solution

Solving time : 0.030 (sec)
Leaf size : 5

dsolve(x*sin(x)*diff(y(x),x)^2 = 0, 
       y(x),singsol=all)
 

\[ y = c_1 \]

1.45.4 Mathematica DSolve solution

Solving time : 0.003 (sec)
Leaf size : 7

DSolve[{x*Sin[x]*D[y[x],x]^2==0,{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 \]