Internal problem ID [2314]
Internal file name [OUTPUT/2314_Tuesday_February_27_2024_08_25_57_AM_5814105/index.tex]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 37, page 171
Problem number: 1.
ODE order: 1.
ODE degree: 2.
The type(s) of ODE detected by this program : "exact", "linear", "separable", "homogeneousTypeD2", "first_order_ode_lie_symmetry_lookup"
Maple gives the following as the ode type
[_separable]
\[ \boxed {4 y^{2}-{y^{\prime }}^{2} x^{2}=0} \] The ode \begin {align*} 4 y^{2}-{y^{\prime }}^{2} x^{2} = 0 \end {align*}
is factored to \begin {align*} \left (-y^{\prime } x +2 y\right ) \left (y^{\prime } x +2 y\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} -y^{\prime } x +2 y = 0\tag {1} \\ y^{\prime } x +2 y = 0\tag {2} \\ \end {align*}
Each of the above equations is now solved.
Solving ODE (1) In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {2 y}{x} \end {align*}
Where \(f(x)=\frac {2}{x}\) and \(g(y)=y\). Integrating both sides gives \begin {align*} \frac {1}{y} \,dy &= \frac {2}{x} \,d x\\ \int { \frac {1}{y} \,dy} &= \int {\frac {2}{x} \,d x}\\ \ln \left (y \right )&=2 \ln \left (x \right )+c_{1}\\ y&={\mathrm e}^{2 \ln \left (x \right )+c_{1}}\\ &=c_{1} x^{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} x^{2} \\ \end{align*}
Verification of solutions
\[ y = c_{1} x^{2} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} x^{2} \\ \end{align*}
Verification of solutions
\[ y = c_{1} x^{2} \] Verified OK.
Solving ODE (2) In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= -\frac {2 y}{x} \end {align*}
Where \(f(x)=-\frac {2}{x}\) and \(g(y)=y\). Integrating both sides gives \begin {align*} \frac {1}{y} \,dy &= -\frac {2}{x} \,d x\\ \int { \frac {1}{y} \,dy} &= \int {-\frac {2}{x} \,d x}\\ \ln \left (y \right )&=-2 \ln \left (x \right )+c_{2}\\ y&={\mathrm e}^{-2 \ln \left (x \right )+c_{2}}\\ &=\frac {c_{2}}{x^{2}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{2}}{x^{2}} \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{2}}{x^{2}} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{2}}{x^{2}} \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{2}}{x^{2}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 4 y^{2}-{y^{\prime }}^{2} x^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=-\frac {2 y}{x}, y^{\prime }=\frac {2 y}{x}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {2 y}{x} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=-\frac {2}{x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int -\frac {2}{x}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=-2 \ln \left (x \right )+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {{\mathrm e}^{c_{1}}}{x^{2}} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {2 y}{x} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=\frac {2}{x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int \frac {2}{x}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=2 \ln \left (x \right )+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=x^{2} {\mathrm e}^{c_{1}} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=x^{2} {\mathrm e}^{c_{1}}, y=\frac {{\mathrm e}^{c_{1}}}{x^{2}}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 17
dsolve(4*y(x)^2=diff(y(x),x)^2*x^2,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= c_{1} x^{2} \\ y \left (x \right ) &= \frac {c_{1}}{x^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.038 (sec). Leaf size: 26
DSolve[4*y[x]^2==y'[x]^2*x^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1}{x^2} \\ y(x)\to c_1 x^2 \\ y(x)\to 0 \\ \end{align*}