Internal problem ID [15088]
Internal file name [OUTPUT/15089_Sunday_April_21_2024_01_29_33_PM_83060937/index.tex]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Section 8. First order not solved for the derivative. Exercises page 67
Problem number: 201.
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]
\[ \boxed {{y^{\prime }}^{2}-2 y^{\prime } x=8 x^{2}} \] The ode \begin {align*} {y^{\prime }}^{2}-2 y^{\prime } x = 8 x^{2} \end {align*}
is factored to \begin {align*} \left (y^{\prime }+2 x \right ) \left (y^{\prime }-4 x \right ) = 0 \end {align*}
Which gives the following equations \begin {align*} y^{\prime }+2 x = 0\tag {1} \\ y^{\prime }-4 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 { 4 x\,\mathop {\mathrm {d}x}}\\ &= 2 x^{2}+c_{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= 2 x^{2}+c_{2} \\ \end{align*}
Verification of solutions
\[ y = 2 x^{2}+c_{2} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= 2 x^{2}+c_{2} \\ \end{align*}
Verification of solutions
\[ y = 2 x^{2}+c_{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{2}-2 y^{\prime } x =8 x^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=-2 x , y^{\prime }=4 x \right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-2 x \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int -2 x d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-x^{2}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-x^{2}+c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=4 x \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int 4 x d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=2 x^{2}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=2 x^{2}+c_{1} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=-x^{2}+c_{1} , y=2 x^{2}+c_{1} \right \} \end {array} \]
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: 21
dsolve(diff(y(x),x)^2-2*x*diff(y(x),x)-8*x^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 2 x^{2}+c_{1} \\ y \left (x \right ) &= -x^{2}+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 25
DSolve[y'[x]^2-2*x*y'[x]-8*x^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -x^2+c_1 \\ y(x)\to 2 x^2+c_1 \\ \end{align*}