Internal problem ID [566]
Internal file name [OUTPUT/566_Sunday_June_05_2022_01_44_51_AM_23911975/index.tex
]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Section 2.6. Page 100
Problem number: 30.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "exact", "differentialType"
Maple gives the following as the ode type
[_rational]
\[ \boxed {\frac {6}{y}+\left (\frac {x^{2}}{y}+\frac {3 y}{x}\right ) y^{\prime }=-3 x} \]
Writing the ode as \begin {align*} y^{\prime }&=\frac {-3 x -\frac {6}{y}}{\frac {x^{2}}{y}+\frac {3 y}{x}}\tag {1} \end {align*}
Which becomes \begin {align*} \left (3 y^{2}\right ) dy &= \left (-x^{3}\right ) dy + \left (-3 x \left (y x +2\right )\right ) dx\tag {2} \end {align*}
But the RHS is complete differential because \begin {align*} \left (-x^{3}\right ) dy + \left (-3 x \left (y x +2\right )\right ) dx &= d\left (-y \,x^{3}-3 x^{2}\right ) \end {align*}
Hence (2) becomes \begin {align*} \left (3 y^{2}\right ) dy &= d\left (-y \,x^{3}-3 x^{2}\right ) \end {align*}
Integrating both sides gives gives these solutions \begin {align*} y&=\frac {\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{3}}{\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}+c_{1}\\ y&=-\frac {\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{12}+\frac {x^{3}}{\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{3}}{\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}+c_{1}\\ y&=-\frac {\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{12}+\frac {x^{3}}{\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{3}}{\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{3}}{\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}+c_{1} \\ \tag{2} y &= -\frac {\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{12}+\frac {x^{3}}{\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{3}}{\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}+c_{1} \\ \tag{3} y &= -\frac {\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{12}+\frac {x^{3}}{\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{3}}{\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \frac {\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{3}}{\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}+c_{1} \] Verified OK.
\[ y = -\frac {\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{12}+\frac {x^{3}}{\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{3}}{\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}+c_{1} \] Verified OK.
\[ y = -\frac {\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{12}+\frac {x^{3}}{\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{3}}{\left (-324 x^{2}+108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}-486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}+c_{1} \] Verified OK.
Entering Exact first order ODE solver. (Form one type)
To solve an ode of the form\begin {equation} M\left ( x,y\right ) +N\left ( x,y\right ) \frac {dy}{dx}=0\tag {A} \end {equation} We assume there exists a function \(\phi \left ( x,y\right ) =c\) where \(c\) is constant, that satisfies the ode. Taking derivative of \(\phi \) w.r.t. \(x\) gives\[ \frac {d}{dx}\phi \left ( x,y\right ) =0 \] Hence\begin {equation} \frac {\partial \phi }{\partial x}+\frac {\partial \phi }{\partial y}\frac {dy}{dx}=0\tag {B} \end {equation} Comparing (A,B) shows that\begin {align*} \frac {\partial \phi }{\partial x} & =M\\ \frac {\partial \phi }{\partial y} & =N \end {align*}
But since \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) then for the above to be valid, we require that\[ \frac {\partial M}{\partial y}=\frac {\partial N}{\partial x}\] If the above condition is satisfied, then the original ode is called exact. We still need to determine \(\phi \left ( x,y\right ) \) but at least we know now that we can do that since the condition \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) is satisfied. If this condition is not satisfied then this method will not work and we have to now look for an integrating factor to force this condition, which might or might not exist. The first step is to write the ODE in standard form to check for exactness, which is \[ M(x,y) \mathop {\mathrm {d}x}+ N(x,y) \mathop {\mathrm {d}y}=0 \tag {1A} \] Therefore \begin {align*} \left (x^{3}+3 y^{2}\right )\mathop {\mathrm {d}y} &= \left (-3 x \left (y x +2\right )\right )\mathop {\mathrm {d}x}\\ \left (3 x \left (y x +2\right )\right )\mathop {\mathrm {d}x} + \left (x^{3}+3 y^{2}\right )\mathop {\mathrm {d}y} &= 0 \tag {2A} \end {align*}
Comparing (1A) and (2A) shows that \begin {align*} M(x,y) &= 3 x \left (y x +2\right )\\ N(x,y) &= x^{3}+3 y^{2} \end {align*}
The next step is to determine if the ODE is is exact or not. The ODE is exact when the following condition is satisfied \[ \frac {\partial M}{\partial y} = \frac {\partial N}{\partial x} \] Using result found above gives \begin {align*} \frac {\partial M}{\partial y} &= \frac {\partial }{\partial y} \left (3 x \left (y x +2\right )\right )\\ &= 3 x^{2} \end {align*}
And \begin {align*} \frac {\partial N}{\partial x} &= \frac {\partial }{\partial x} \left (x^{3}+3 y^{2}\right )\\ &= 3 x^{2} \end {align*}
Since \(\frac {\partial M}{\partial y}= \frac {\partial N}{\partial x}\), then the ODE is exact The following equations are now set up to solve for the function \(\phi \left (x,y\right )\) \begin {align*} \frac {\partial \phi }{\partial x } &= M\tag {1} \\ \frac {\partial \phi }{\partial y } &= N\tag {2} \end {align*}
Integrating (1) w.r.t. \(x\) gives \begin{align*} \int \frac {\partial \phi }{\partial x} \mathop {\mathrm {d}x} &= \int M\mathop {\mathrm {d}x} \\ \int \frac {\partial \phi }{\partial x} \mathop {\mathrm {d}x} &= \int 3 x \left (y x +2\right )\mathop {\mathrm {d}x} \\ \tag{3} \phi &= y \,x^{3}+3 x^{2}+ f(y) \\ \end{align*} Where \(f(y)\) is used for the constant of integration since \(\phi \) is a function of both \(x\) and \(y\). Taking derivative of equation (3) w.r.t \(y\) gives \begin{equation} \tag{4} \frac {\partial \phi }{\partial y} = x^{3}+f'(y) \end{equation} But equation (2) says that \(\frac {\partial \phi }{\partial y} = x^{3}+3 y^{2}\). Therefore equation (4) becomes \begin{equation} \tag{5} x^{3}+3 y^{2} = x^{3}+f'(y) \end{equation} Solving equation (5) for \( f'(y)\) gives \[ f'(y) = 3 y^{2} \] Integrating the above w.r.t \(y\) gives \begin{align*} \int f'(y) \mathop {\mathrm {d}y} &= \int \left ( 3 y^{2}\right ) \mathop {\mathrm {d}y} \\ f(y) &= y^{3}+ c_{1} \\ \end{align*} Where \(c_{1}\) is constant of integration. Substituting result found above for \(f(y)\) into equation (3) gives \(\phi \) \[ \phi = y \,x^{3}+y^{3}+3 x^{2}+ c_{1} \] But since \(\phi \) itself is a constant function, then let \(\phi =c_{2}\) where \(c_{2}\) is new constant and combining \(c_{1}\) and \(c_{2}\) constants into new constant \(c_{1}\) gives the solution as \[ c_{1} = y \,x^{3}+y^{3}+3 x^{2} \]
The solution(s) found are the following \begin{align*} \tag{1} y x^{3}+y^{3}+3 x^{2} &= c_{1} \\ \end{align*}
Verification of solutions
\[ y x^{3}+y^{3}+3 x^{2} = c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {6}{y}+\left (\frac {x^{2}}{y}+\frac {3 y}{x}\right ) y^{\prime }=-3 x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-3 x -\frac {6}{y}}{\frac {x^{2}}{y}+\frac {3 y}{x}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact <- exact successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 326
dsolve(3*x+6/y(x)+(x^2/y(x)+3*y(x)/x)*diff(y(x),x) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {-12 x^{3}+\left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {2}{3}}}{6 \left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{12}-\frac {x^{3} \left (i \sqrt {3}-1\right )}{\left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {12 i \sqrt {3}\, x^{3}+i \sqrt {3}\, \left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {2}{3}}+12 x^{3}-\left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {2}{3}}}{12 \left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.558 (sec). Leaf size: 331
DSolve[3*x+6/y[x]+(x^2/y[x]+3*y[x]/x)*y'[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{-81 x^2+\sqrt {108 x^9+729 \left (-3 x^2+c_1\right ){}^2}+27 c_1}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} x^3}{\sqrt [3]{-81 x^2+\sqrt {108 x^9+729 \left (-3 x^2+c_1\right ){}^2}+27 c_1}} \\ y(x)\to \frac {\left (-1+i \sqrt {3}\right ) \sqrt [3]{-81 x^2+\sqrt {108 x^9+729 \left (-3 x^2+c_1\right ){}^2}+27 c_1}}{6 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) x^3}{2^{2/3} \sqrt [3]{-81 x^2+\sqrt {108 x^9+729 \left (-3 x^2+c_1\right ){}^2}+27 c_1}} \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) x^3}{2^{2/3} \sqrt [3]{-81 x^2+\sqrt {108 x^9+729 \left (-3 x^2+c_1\right ){}^2}+27 c_1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-81 x^2+\sqrt {108 x^9+729 \left (-3 x^2+c_1\right ){}^2}+27 c_1}}{6 \sqrt [3]{2}} \\ \end{align*}