Internal problem ID [3484]
Internal file name [OUTPUT/2977_Sunday_June_05_2022_08_48_41_AM_61921181/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 8
Problem number: 228.
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]
\[ \boxed {\left (a +x \right ) y^{\prime }=b x} \]
Integrating both sides gives \begin {align*} y &= \int { \frac {b x}{a +x}\,\mathop {\mathrm {d}x}}\\ &= b \left (x -a \ln \left (a +x \right )\right )+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= b \left (x -a \ln \left (a +x \right )\right )+c_{1} \\ \end{align*}
Verification of solutions
\[ y = b \left (x -a \ln \left (a +x \right )\right )+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (a +x \right ) y^{\prime }=b x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {b x}{a +x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {b x}{a +x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=b \left (x -a \ln \left (a +x \right )\right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-a b \ln \left (a +x \right )+b x +c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 17
dsolve((a+x)*diff(y(x),x) = b*x,y(x), singsol=all)
\[ y \left (x \right ) = -\ln \left (x +a \right ) a b +b x +c_{1} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 19
DSolve[(a+x) y'[x]==b x,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -a b \log (a+x)+b x+c_1 \]