48.1.13 problem Example 3.14

Internal problem ID [7514]
Book : THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS. K.T. CHAU, CRC Press. Boca Raton, FL. 2018
Section : Chapter 3. Ordinary Differential Equations. Section 3.2 FIRST ORDER ODE. Page 114
Problem number : Example 3.14
Date solved : Tuesday, January 28, 2025 at 03:10:51 PM
CAS classification : [_dAlembert]

\begin{align*} y^{\prime } y+x&=a {y^{\prime }}^{2} \end{align*}

✓ Solution by Maple

Time used: 0.035 (sec). Leaf size: 246

dsolve(x+y(x)*diff(y(x),x)=a*(diff(y(x),x))^2,y(x), singsol=all)
 

\begin{align*} \frac {-\frac {\sqrt {2}\, \left (y+\sqrt {4 a x +y^{2}}\right ) \operatorname {arcsinh}\left (\frac {y+\sqrt {4 a x +y^{2}}}{2 a}\right )}{2}+x \sqrt {\frac {y \sqrt {4 a x +y^{2}}+2 a^{2}+2 a x +y^{2}}{a^{2}}}+c_{1} y+c_{1} \sqrt {4 a x +y^{2}}}{\sqrt {\frac {y \sqrt {4 a x +y^{2}}+y^{2}+2 a \left (a +x \right )}{a^{2}}}} &= 0 \\ \frac {x \sqrt {\frac {-2 y \sqrt {4 a x +y^{2}}+2 y^{2}+4 a \left (a +x \right )}{a^{2}}}-\left (y-\sqrt {4 a x +y^{2}}\right ) \left (-\operatorname {arcsinh}\left (\frac {-y+\sqrt {4 a x +y^{2}}}{2 a}\right )+c_{1} \right )}{\sqrt {\frac {-2 y \sqrt {4 a x +y^{2}}+2 y^{2}+4 a \left (a +x \right )}{a^{2}}}} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.517 (sec). Leaf size: 57

DSolve[x+y[x]*D[y[x],x]==a*(D[y[x],x])^2,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\left \{x=\frac {a K[1] \text {arcsinh}(K[1])}{\sqrt {K[1]^2+1}}+\frac {c_1 K[1]}{\sqrt {K[1]^2+1}},y(x)=a K[1]-\frac {x}{K[1]}\right \},\{y(x),K[1]\}\right ] \]