Internal problem ID [5095]
Internal file name [OUTPUT/4588_Sunday_June_05_2022_03_01_24_PM_29063218/index.tex]
Book: Engineering Mathematics. By K. A. Stroud. 5th edition. Industrial press Inc. NY.
2001
Section: Program 24. First order differential equations. Further problems 24. page
1068
Problem number: 9.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]
\[ \boxed {y+\left (4 y+3 x \right ) y^{\prime }=3 x} \]
In canonical form, the ODE is \begin {align*} y' &= F(x,y)\\ &= -\frac {y -3 x}{4 y +3 x}\tag {1} \end {align*}
An ode of the form \(y' = \frac {M(x,y)}{N(x,y)}\) is called homogeneous if the functions \(M(x,y)\) and \(N(x,y)\) are both homogeneous functions and of the same order. Recall that a function \(f(x,y)\) is homogeneous of order \(n\) if \[ f(t^n x, t^n y)= t^n f(x,y) \] In this case, it can be seen that both \(M=-y +3 x\) and \(N=4 y +3 x\) are both homogeneous and of the same order \(n=1\). Therefore this is a homogeneous ode. Since this ode is homogeneous, it is converted to separable ODE using the substitution \(u=\frac {y}{x}\), or \(y=ux\). Hence \[ \frac { \mathop {\mathrm {d}y}}{\mathop {\mathrm {d}x}}= \frac { \mathop {\mathrm {d}u}}{\mathop {\mathrm {d}x}}x + u \] Applying the transformation \(y=ux\) to the above ODE in (1) gives \begin {align*} \frac { \mathop {\mathrm {d}u}}{\mathop {\mathrm {d}x}}x + u &= \frac {-u +3}{4 u +3}\\ \frac { \mathop {\mathrm {d}u}}{\mathop {\mathrm {d}x}} &= \frac {\frac {-u \left (x \right )+3}{4 u \left (x \right )+3}-u \left (x \right )}{x} \end {align*}
Or \[ u^{\prime }\left (x \right )-\frac {\frac {-u \left (x \right )+3}{4 u \left (x \right )+3}-u \left (x \right )}{x} = 0 \] Or \[ 4 u^{\prime }\left (x \right ) x u \left (x \right )+3 u^{\prime }\left (x \right ) x +4 u \left (x \right )^{2}+4 u \left (x \right )-3 = 0 \] Or \[ -3+x \left (4 u \left (x \right )+3\right ) u^{\prime }\left (x \right )+4 u \left (x \right )^{2}+4 u \left (x \right ) = 0 \] Which is now solved as separable in \(u \left (x \right )\). Which is now solved in \(u \left (x \right )\). In canonical form the ODE is \begin {align*} u' &= F(x,u)\\ &= f( x) g(u)\\ &= -\frac {4 u^{2}+4 u -3}{x \left (4 u +3\right )} \end {align*}
Where \(f(x)=-\frac {1}{x}\) and \(g(u)=\frac {4 u^{2}+4 u -3}{4 u +3}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {4 u^{2}+4 u -3}{4 u +3}} \,du &= -\frac {1}{x} \,d x \\ \int { \frac {1}{\frac {4 u^{2}+4 u -3}{4 u +3}} \,du} &= \int {-\frac {1}{x} \,d x} \\ \frac {5 \ln \left (u -\frac {1}{2}\right )}{8}+\frac {3 \ln \left (u +\frac {3}{2}\right )}{8}&=-\ln \left (x \right )+c_{2} \\ \end{align*} The above can be written as \begin {align*} \frac {5 \ln \left (u -\frac {1}{2}\right )+3 \ln \left (u +\frac {3}{2}\right )}{8} &= -\ln \left (x \right )+c_{2}\\ 5 \ln \left (u -\frac {1}{2}\right )+3 \ln \left (u +\frac {3}{2}\right ) &= \left (8\right ) \left (-\ln \left (x \right )+c_{2}\right ) \\ &= -8 \ln \left (x \right )+8 c_{2} \end {align*}
Raising both side to exponential gives \begin {align*} {\mathrm e}^{5 \ln \left (u -\frac {1}{2}\right )+3 \ln \left (u +\frac {3}{2}\right )} &= {\mathrm e}^{-8 \ln \left (x \right )+8 c_{2}} \end {align*}
Which simplifies to \begin {align*} \frac {\left (2 u -1\right )^{5} \left (2 u +3\right )^{3}}{256} &= \frac {8 c_{2}}{x^{8}}\\ &= \frac {c_{3}}{x^{8}} \end {align*}
Which simplifies to \[ u \left (x \right ) = \frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+4 \textit {\_Z}^{7}-8 \textit {\_Z}^{6}-28 \textit {\_Z}^{5}+50 \textit {\_Z}^{4}+44 \textit {\_Z}^{3}-\frac {256 c_{3} {\mathrm e}^{8 c_{2}}}{x^{8}}-144 \textit {\_Z}^{2}+108 \textit {\_Z} -27\right )}{2} \] Now \(u\) in the above solution is replaced back by \(y\) using \(u=\frac {y}{x}\) which results in the solution \[ y = \frac {x \operatorname {RootOf}\left (\textit {\_Z}^{8} x^{8}+4 \textit {\_Z}^{7} x^{8}-8 \textit {\_Z}^{6} x^{8}-28 \textit {\_Z}^{5} x^{8}+50 \textit {\_Z}^{4} x^{8}+44 \textit {\_Z}^{3} x^{8}-144 \textit {\_Z}^{2} x^{8}-256 c_{3} {\mathrm e}^{8 c_{2}}+108 \textit {\_Z} \,x^{8}-27 x^{8}\right )}{2} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x \operatorname {RootOf}\left (\textit {\_Z}^{8} x^{8}+4 \textit {\_Z}^{7} x^{8}-8 \textit {\_Z}^{6} x^{8}-28 \textit {\_Z}^{5} x^{8}+50 \textit {\_Z}^{4} x^{8}+44 \textit {\_Z}^{3} x^{8}-144 \textit {\_Z}^{2} x^{8}-256 c_{3} {\mathrm e}^{8 c_{2}}+108 \textit {\_Z} \,x^{8}-27 x^{8}\right )}{2} \\ \end{align*}
Verification of solutions
\[ y = \frac {x \operatorname {RootOf}\left (\textit {\_Z}^{8} x^{8}+4 \textit {\_Z}^{7} x^{8}-8 \textit {\_Z}^{6} x^{8}-28 \textit {\_Z}^{5} x^{8}+50 \textit {\_Z}^{4} x^{8}+44 \textit {\_Z}^{3} x^{8}-144 \textit {\_Z}^{2} x^{8}-256 c_{3} {\mathrm e}^{8 c_{2}}+108 \textit {\_Z} \,x^{8}-27 x^{8}\right )}{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y+\left (4 y+3 x \right ) y^{\prime }=3 x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-y+3 x}{4 y+3 x} \end {array} \]
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 homogeneous D <- homogeneous successful`
✓ Solution by Maple
Time used: 0.36 (sec). Leaf size: 278
dsolve(y(x)-3*x+(4*y(x)+3*x)*diff(y(x),x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-3 x^{8} c_{1} \operatorname {RootOf}\left (\textit {\_Z}^{64} c_{1} x^{8}+12 \textit {\_Z}^{56} c_{1} x^{8}+48 \textit {\_Z}^{48} c_{1} x^{8}+64 \textit {\_Z}^{40} c_{1} x^{8}-1\right )^{56}-24 x^{8} c_{1} \operatorname {RootOf}\left (\textit {\_Z}^{64} c_{1} x^{8}+12 \textit {\_Z}^{56} c_{1} x^{8}+48 \textit {\_Z}^{48} c_{1} x^{8}+64 \textit {\_Z}^{40} c_{1} x^{8}-1\right )^{48}-48 x^{8} c_{1} \operatorname {RootOf}\left (\textit {\_Z}^{64} c_{1} x^{8}+12 \textit {\_Z}^{56} c_{1} x^{8}+48 \textit {\_Z}^{48} c_{1} x^{8}+64 \textit {\_Z}^{40} c_{1} x^{8}-1\right )^{40}+1}{2 c_{1} x^{7} \operatorname {RootOf}\left (\textit {\_Z}^{64} c_{1} x^{8}+12 \textit {\_Z}^{56} c_{1} x^{8}+48 \textit {\_Z}^{48} c_{1} x^{8}+64 \textit {\_Z}^{40} c_{1} x^{8}-1\right )^{40} \left (\operatorname {RootOf}\left (\textit {\_Z}^{64} c_{1} x^{8}+12 \textit {\_Z}^{56} c_{1} x^{8}+48 \textit {\_Z}^{48} c_{1} x^{8}+64 \textit {\_Z}^{40} c_{1} x^{8}-1\right )^{16}+8 \operatorname {RootOf}\left (\textit {\_Z}^{64} c_{1} x^{8}+12 \textit {\_Z}^{56} c_{1} x^{8}+48 \textit {\_Z}^{48} c_{1} x^{8}+64 \textit {\_Z}^{40} c_{1} x^{8}-1\right )^{8}+16\right )} \]
✓ Solution by Mathematica
Time used: 5.296 (sec). Leaf size: 673
DSolve[y[x]-3*x+(4*y[x]+3*x)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {Root}\left [256 \text {$\#$1}^8+512 \text {$\#$1}^7 x-512 \text {$\#$1}^6 x^2-896 \text {$\#$1}^5 x^3+800 \text {$\#$1}^4 x^4+352 \text {$\#$1}^3 x^5-576 \text {$\#$1}^2 x^6+216 \text {$\#$1} x^7-27 x^8+e^{8 c_1}\&,1\right ] \\ y(x)\to \text {Root}\left [256 \text {$\#$1}^8+512 \text {$\#$1}^7 x-512 \text {$\#$1}^6 x^2-896 \text {$\#$1}^5 x^3+800 \text {$\#$1}^4 x^4+352 \text {$\#$1}^3 x^5-576 \text {$\#$1}^2 x^6+216 \text {$\#$1} x^7-27 x^8+e^{8 c_1}\&,2\right ] \\ y(x)\to \text {Root}\left [256 \text {$\#$1}^8+512 \text {$\#$1}^7 x-512 \text {$\#$1}^6 x^2-896 \text {$\#$1}^5 x^3+800 \text {$\#$1}^4 x^4+352 \text {$\#$1}^3 x^5-576 \text {$\#$1}^2 x^6+216 \text {$\#$1} x^7-27 x^8+e^{8 c_1}\&,3\right ] \\ y(x)\to \text {Root}\left [256 \text {$\#$1}^8+512 \text {$\#$1}^7 x-512 \text {$\#$1}^6 x^2-896 \text {$\#$1}^5 x^3+800 \text {$\#$1}^4 x^4+352 \text {$\#$1}^3 x^5-576 \text {$\#$1}^2 x^6+216 \text {$\#$1} x^7-27 x^8+e^{8 c_1}\&,4\right ] \\ y(x)\to \text {Root}\left [256 \text {$\#$1}^8+512 \text {$\#$1}^7 x-512 \text {$\#$1}^6 x^2-896 \text {$\#$1}^5 x^3+800 \text {$\#$1}^4 x^4+352 \text {$\#$1}^3 x^5-576 \text {$\#$1}^2 x^6+216 \text {$\#$1} x^7-27 x^8+e^{8 c_1}\&,5\right ] \\ y(x)\to \text {Root}\left [256 \text {$\#$1}^8+512 \text {$\#$1}^7 x-512 \text {$\#$1}^6 x^2-896 \text {$\#$1}^5 x^3+800 \text {$\#$1}^4 x^4+352 \text {$\#$1}^3 x^5-576 \text {$\#$1}^2 x^6+216 \text {$\#$1} x^7-27 x^8+e^{8 c_1}\&,6\right ] \\ y(x)\to \text {Root}\left [256 \text {$\#$1}^8+512 \text {$\#$1}^7 x-512 \text {$\#$1}^6 x^2-896 \text {$\#$1}^5 x^3+800 \text {$\#$1}^4 x^4+352 \text {$\#$1}^3 x^5-576 \text {$\#$1}^2 x^6+216 \text {$\#$1} x^7-27 x^8+e^{8 c_1}\&,7\right ] \\ y(x)\to \text {Root}\left [256 \text {$\#$1}^8+512 \text {$\#$1}^7 x-512 \text {$\#$1}^6 x^2-896 \text {$\#$1}^5 x^3+800 \text {$\#$1}^4 x^4+352 \text {$\#$1}^3 x^5-576 \text {$\#$1}^2 x^6+216 \text {$\#$1} x^7-27 x^8+e^{8 c_1}\&,8\right ] \\ \end{align*}