Internal problem ID [5320]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 6. Equations of first order and first degree (Linear equations). Supplemetary
problems. Page 39
Problem number: 23 (c).
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Bernoulli]
\[ \boxed {x y^{3}-y^{3}+3 x y^{2} y^{\prime }=x^{2} {\mathrm e}^{x}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 140
dsolve((x*y(x)^3-y(x)^3-x^2*exp(x))+(3*x*y(x)^2)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y = \frac {{\mathrm e}^{-x} {\left (\left (4 \,{\mathrm e}^{2 x}+8 c_{1} \right ) x \,{\mathrm e}^{2 x}\right )}^{\frac {1}{3}}}{2} y = -\frac {{\mathrm e}^{-x} {\left (\left (4 \,{\mathrm e}^{2 x}+8 c_{1} \right ) x \,{\mathrm e}^{2 x}\right )}^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, {\mathrm e}^{-x} {\left (\left (4 \,{\mathrm e}^{2 x}+8 c_{1} \right ) x \,{\mathrm e}^{2 x}\right )}^{\frac {1}{3}}}{4} y = -\frac {{\mathrm e}^{-x} {\left (\left (4 \,{\mathrm e}^{2 x}+8 c_{1} \right ) x \,{\mathrm e}^{2 x}\right )}^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, {\mathrm e}^{-x} {\left (\left (4 \,{\mathrm e}^{2 x}+8 c_{1} \right ) x \,{\mathrm e}^{2 x}\right )}^{\frac {1}{3}}}{4} \end{align*}
✓ Solution by Mathematica
Time used: 0.854 (sec). Leaf size: 117
DSolve[(x*y[x]^3-y[x]^3-x^2*Exp[x])+(3*x*y[x]^2)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt [3]{-\frac {1}{2}} e^{-x/3} \sqrt [3]{x} \sqrt [3]{e^{2 x}+2 c_1} y(x)\to \frac {e^{-x/3} \sqrt [3]{x} \sqrt [3]{e^{2 x}+2 c_1}}{\sqrt [3]{2}} y(x)\to \frac {(-1)^{2/3} e^{-x/3} \sqrt [3]{x} \sqrt [3]{e^{2 x}+2 c_1}}{\sqrt [3]{2}} \end{align*}