Internal problem ID [7065]
Book: Own collection of miscellaneous problems
Section: section 1.0
Problem number: 21.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _Chini]
\[ \boxed {y^{\prime }-\sqrt {y}=x} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 69
dsolve(diff(y(x),x)=sqrt(y(x))+x,y(x), singsol=all)
\[ \frac {4 \,\operatorname {arctanh}\left (\sqrt {\frac {y \left (x \right )}{x^{2}}}\right )}{3}-\frac {2 \,\operatorname {arctanh}\left (2 \sqrt {\frac {y \left (x \right )}{x^{2}}}\right )}{3}-\frac {\ln \left (\frac {-x^{2}+4 y \left (x \right )}{x^{2}}\right )}{3}-\frac {2 \ln \left (2\right )}{3}-\frac {2 \ln \left (\frac {y \left (x \right )-x^{2}}{x^{2}}\right )}{3}-2 \ln \left (x \right )+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 47.265 (sec). Leaf size: 716
DSolve[y'[x]==Sqrt[y[x]]+x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{4} \left (3 x^2+\frac {e^{3 c_1} x \left (8+e^{3 c_1} x^3\right )}{\sqrt [3]{-e^{18 c_1} x^6+20 e^{15 c_1} x^3+8 \sqrt {-e^{24 c_1} \left (-1+e^{3 c_1} x^3\right ){}^3}+8 e^{12 c_1}}}+e^{-6 c_1} \sqrt [3]{-e^{18 c_1} x^6+20 e^{15 c_1} x^3+8 \sqrt {-e^{24 c_1} \left (-1+e^{3 c_1} x^3\right ){}^3}+8 e^{12 c_1}}\right ) \\ y(x)\to \frac {1}{72} \left (54 x^2-\frac {9 i \left (\sqrt {3}-i\right ) e^{3 c_1} x \left (8+e^{3 c_1} x^3\right )}{\sqrt [3]{-e^{18 c_1} x^6+20 e^{15 c_1} x^3+8 \sqrt {-e^{24 c_1} \left (-1+e^{3 c_1} x^3\right ){}^3}+8 e^{12 c_1}}}+9 i \left (\sqrt {3}+i\right ) e^{-6 c_1} \sqrt [3]{-e^{18 c_1} x^6+20 e^{15 c_1} x^3+8 \sqrt {-e^{24 c_1} \left (-1+e^{3 c_1} x^3\right ){}^3}+8 e^{12 c_1}}\right ) \\ y(x)\to \frac {1}{72} \left (54 x^2+\frac {9 i \left (\sqrt {3}+i\right ) e^{3 c_1} x \left (8+e^{3 c_1} x^3\right )}{\sqrt [3]{-e^{18 c_1} x^6+20 e^{15 c_1} x^3+8 \sqrt {-e^{24 c_1} \left (-1+e^{3 c_1} x^3\right ){}^3}+8 e^{12 c_1}}}-9 \left (1+i \sqrt {3}\right ) e^{-6 c_1} \sqrt [3]{-e^{18 c_1} x^6+20 e^{15 c_1} x^3+8 \sqrt {-e^{24 c_1} \left (-1+e^{3 c_1} x^3\right ){}^3}+8 e^{12 c_1}}\right ) \\ y(x)\to \frac {-\left (-x^6\right )^{2/3}+3 x^4+\sqrt [3]{-x^6} x^2}{4 x^2} \\ y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (-x^6\right )^{2/3}+6 x^4+i \left (\sqrt {3}+i\right ) \sqrt [3]{-x^6} x^2}{8 x^2} \\ y(x)\to \frac {1}{8} x^2 \left (\frac {\left (1+i \sqrt {3}\right ) x^4}{\left (-x^6\right )^{2/3}}+\frac {i \left (\sqrt {3}+i\right ) x^2}{\sqrt [3]{-x^6}}+6\right ) \\ \end{align*}