Internal problem ID [5848]
Internal file name [OUTPUT/5096_Sunday_June_05_2022_03_24_32_PM_70360323/index.tex
]
Book: THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS.
K.T. CHAU, CRC Press. Boca Raton, FL. 2018
Section: Chapter 3. Ordinary Differential Equations. Section 3.2 FIRST ORDER ODE. Page
114
Problem number: Example 3.16.
ODE order: 1.
ODE degree: 2.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {{y^{\prime }}^{2}=4 x^{2}} \] The ode \begin {align*} {y^{\prime }}^{2} = 4 x^{2} \end {align*}
is factored to \begin {align*} \left (y^{\prime }-2 x \right ) \left (y^{\prime }+2 x \right ) = 0 \end {align*}
Which gives the following equations \begin {align*} y^{\prime }-2 x = 0\tag {1} \\ y^{\prime }+2 x = 0\tag {2} \\ \end {align*}
Each of the above equations is now solved.
Solving ODE (1) Integrating both sides gives \begin {align*} y &= \int { 2 x\,\mathop {\mathrm {d}x}}\\ &= x^{2}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= x^{2}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = x^{2}+c_{1} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= x^{2}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = x^{2}+c_{1} \] Verified OK.
Solving ODE (2) Integrating both sides gives \begin {align*} y &= \int { -2 x\,\mathop {\mathrm {d}x}}\\ &= -x^{2}+c_{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -x^{2}+c_{2} \\ \end{align*}
Verification of solutions
\[ y = -x^{2}+c_{2} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -x^{2}+c_{2} \\ \end{align*}
Verification of solutions
\[ y = -x^{2}+c_{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{2}=4 x^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=-2 x , y^{\prime }=2 x \right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-2 x \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int -2 x d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-x^{2}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-x^{2}+c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=2 x \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int 2 x d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=x^{2}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=x^{2}+c_{1} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=-x^{2}+c_{1} , y=x^{2}+c_{1} \right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 19
dsolve(diff(y(x),x)^2=4*x^2,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= x^{2}+c_{1} \\ y \left (x \right ) &= -x^{2}+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 23
DSolve[(y'[x])^2==4*x^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -x^2+c_1 \\ y(x)\to x^2+c_1 \\ \end{align*}