Internal problem ID [4225]
Internal file name [OUTPUT/3718_Sunday_June_05_2022_10_25_20_AM_90759115/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 33
Problem number: 993.
ODE order: 1.
ODE degree: 2.
The type(s) of ODE detected by this program : "exact", "separable", "differentialType", "homogeneousTypeD2", "first_order_ode_lie_symmetry_lookup"
Maple gives the following as the ode type
[_separable]
\[ \boxed {4 y^{2} {y^{\prime }}^{2}+2 \left (1+3 x \right ) x y y^{\prime }=-3 x^{3}} \] The ode \begin {align*} 4 y^{2} {y^{\prime }}^{2}+2 \left (1+3 x \right ) x y y^{\prime } = -3 x^{3} \end {align*}
is factored to \begin {align*} \left (2 y^{\prime } y+x \right ) \left (2 y^{\prime } y+3 x^{2}\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} 2 y^{\prime } y+x = 0\tag {1} \\ 2 y^{\prime } y+3 x^{2} = 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 {x}{2 y} \end {align*}
Where \(f(x)=-\frac {x}{2}\) and \(g(y)=\frac {1}{y}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {1}{y}} \,dy &= -\frac {x}{2} \,d x \\ \int { \frac {1}{\frac {1}{y}} \,dy} &= \int {-\frac {x}{2} \,d x} \\ \frac {y^{2}}{2}&=-\frac {x^{2}}{4}+c_{1} \\ \end{align*} Which results in \begin{align*} y &= \frac {\sqrt {-2 x^{2}+8 c_{1}}}{2} \\ y &= -\frac {\sqrt {-2 x^{2}+8 c_{1}}}{2} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\sqrt {-2 x^{2}+8 c_{1}}}{2} \\ \tag{2} y &= -\frac {\sqrt {-2 x^{2}+8 c_{1}}}{2} \\ \end{align*}
Verification of solutions
\[ y = \frac {\sqrt {-2 x^{2}+8 c_{1}}}{2} \] Verified OK.
\[ y = -\frac {\sqrt {-2 x^{2}+8 c_{1}}}{2} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\sqrt {-2 x^{2}+8 c_{1}}}{2} \\ \tag{2} y &= -\frac {\sqrt {-2 x^{2}+8 c_{1}}}{2} \\ \end{align*}
Verification of solutions
\[ y = \frac {\sqrt {-2 x^{2}+8 c_{1}}}{2} \] Verified OK.
\[ y = -\frac {\sqrt {-2 x^{2}+8 c_{1}}}{2} \] Verified OK.
Solving ODE (2) In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= -\frac {3 x^{2}}{2 y} \end {align*}
Where \(f(x)=-\frac {3 x^{2}}{2}\) and \(g(y)=\frac {1}{y}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {1}{y}} \,dy &= -\frac {3 x^{2}}{2} \,d x \\ \int { \frac {1}{\frac {1}{y}} \,dy} &= \int {-\frac {3 x^{2}}{2} \,d x} \\ \frac {y^{2}}{2}&=-\frac {x^{3}}{2}+c_{2} \\ \end{align*} Which results in \begin{align*} y &= \sqrt {-x^{3}+2 c_{2}} \\ y &= -\sqrt {-x^{3}+2 c_{2}} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \sqrt {-x^{3}+2 c_{2}} \\ \tag{2} y &= -\sqrt {-x^{3}+2 c_{2}} \\ \end{align*}
Verification of solutions
\[ y = \sqrt {-x^{3}+2 c_{2}} \] Verified OK.
\[ y = -\sqrt {-x^{3}+2 c_{2}} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \sqrt {-x^{3}+2 c_{2}} \\ \tag{2} y &= -\sqrt {-x^{3}+2 c_{2}} \\ \end{align*}
Verification of solutions
\[ y = \sqrt {-x^{3}+2 c_{2}} \] Verified OK.
\[ y = -\sqrt {-x^{3}+2 c_{2}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 4 y^{2} {y^{\prime }}^{2}+2 \left (1+3 x \right ) x y y^{\prime }=-3 x^{3} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=-\frac {x}{2 y}, y^{\prime }=-\frac {3 x^{2}}{2 y}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {x}{2 y} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime } y=-\frac {x}{2} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime } yd x =\int -\frac {x}{2}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {y^{2}}{2}=-\frac {x^{2}}{4}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=-\frac {\sqrt {-2 x^{2}+8 c_{1}}}{2}, y=\frac {\sqrt {-2 x^{2}+8 c_{1}}}{2}\right \} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {3 x^{2}}{2 y} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime } y=-\frac {3 x^{2}}{2} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime } yd x =\int -\frac {3 x^{2}}{2}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {y^{2}}{2}=-\frac {x^{3}}{2}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\sqrt {-x^{3}+2 c_{1}}, y=-\sqrt {-x^{3}+2 c_{1}}\right \} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\left \{y=\sqrt {-x^{3}+2 c_{1}}, y=-\sqrt {-x^{3}+2 c_{1}}\right \}, \left \{y=-\frac {\sqrt {-2 x^{2}+8 c_{1}}}{2}, y=\frac {\sqrt {-2 x^{2}+8 c_{1}}}{2}\right \}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 59
dsolve(4*y(x)^2*diff(y(x),x)^2+2*(1+3*x)*x*y(x)*diff(y(x),x)+3*x^3 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {\sqrt {-2 x^{2}+4 c_{1}}}{2} \\ y \left (x \right ) &= \frac {\sqrt {-2 x^{2}+4 c_{1}}}{2} \\ y \left (x \right ) &= \sqrt {-x^{3}+c_{1}} \\ y \left (x \right ) &= -\sqrt {-x^{3}+c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.169 (sec). Leaf size: 81
DSolve[4 y[x]^2 (y'[x])^2 +2(1+3 x)x y[x] y'[x]+3 x^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {-x^3+2 c_1} \\ y(x)\to \sqrt {-x^3+2 c_1} \\ y(x)\to -\sqrt {-\frac {x^2}{2}+2 c_1} \\ y(x)\to \sqrt {-\frac {x^2}{2}+2 c_1} \\ \end{align*}