Internal problem ID [15145]
Internal file name [OUTPUT/15146_Tuesday_April_23_2024_04_51_02_PM_9403647/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 12. Miscellaneous problems. Exercises page 93
Problem number: 284.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {x y^{\prime }-y^{\prime }=-x^{2}+3 x} \]
Integrating both sides gives \begin {align*} y &= \int { -\frac {x \left (x -3\right )}{x -1}\,\mathop {\mathrm {d}x}}\\ &= -\frac {x^{2}}{2}+2 x +2 \ln \left (x -1\right )+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {x^{2}}{2}+2 x +2 \ln \left (x -1\right )+c_{1} \\ \end{align*}
Verification of solutions
\[ y = -\frac {x^{2}}{2}+2 x +2 \ln \left (x -1\right )+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y^{\prime }-y^{\prime }=-x^{2}+3 x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-x^{2}+3 x}{x -1} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {-x^{2}+3 x}{x -1}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\frac {x^{2}}{2}+2 x +2 \ln \left (x -1\right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {x^{2}}{2}+2 x +2 \ln \left (x -1\right )+c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 20
dsolve(x^2+x*diff(y(x),x)=3*x+diff(y(x),x),y(x), singsol=all)
\[ y = -\frac {x^{2}}{2}+2 x +2 \ln \left (x -1\right )+c_{1} \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 24
DSolve[x^2+x*y'[x]==3*x+y'[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {1}{2} (x-1)^2+x+2 \log (x-1)+c_1 \]