77.1.81 problem 107 (page 162)

Internal problem ID [17971]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 107 (page 162)
Date solved : Tuesday, January 28, 2025 at 11:19:04 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} y^{\prime \prime }&=\frac {1}{\sqrt {y}} \end{align*}

✓ Solution by Maple

Time used: 0.222 (sec). Leaf size: 65

dsolve(diff(y(x),x$2)=1/sqrt(y(x)),y(x), singsol=all)
 

\begin{align*} \frac {\left (-c_{1} -2 \sqrt {y}\right ) \sqrt {4 \sqrt {y}-c_{1}}}{6}-x -c_{2} &= 0 \\ \frac {\left (c_{1} +2 \sqrt {y}\right ) \sqrt {4 \sqrt {y}-c_{1}}}{6}-x -c_{2} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.051 (sec). Leaf size: 1471

DSolve[D[y[x],{x,2}]==1/Sqrt[y[x]],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {288 c_1 x^2+3 c_1{}^2 \sqrt [3]{10368 x^4+41472 c_2 x^3+720 c_1{}^3 x^2+62208 c_2{}^2 x^2+48 \sqrt {(x+c_2){}^2 \left (36 x^2+72 c_2 x-c_1{}^3+36 c_2{}^2\right ){}^3}+1440 c_2 c_1{}^3 x+41472 c_2{}^3 x-c_1{}^6+720 c_2{}^2 c_1{}^3+10368 c_2{}^4}+\left (10368 x^4+41472 c_2 x^3+720 c_1{}^3 x^2+62208 c_2{}^2 x^2+48 \sqrt {(x+c_2){}^2 \left (36 x^2+72 c_2 x-c_1{}^3+36 c_2{}^2\right ){}^3}+1440 c_2 c_1{}^3 x+41472 c_2{}^3 x-c_1{}^6+720 c_2{}^2 c_1{}^3+10368 c_2{}^4\right ){}^{2/3}+576 c_2 c_1 x+c_1{}^4+288 c_2{}^2 c_1}{16 \sqrt [3]{10368 x^4+41472 c_2 x^3+720 c_1{}^3 x^2+62208 c_2{}^2 x^2+48 \sqrt {(x+c_2){}^2 \left (36 x^2+72 c_2 x-c_1{}^3+36 c_2{}^2\right ){}^3}+1440 c_2 c_1{}^3 x+41472 c_2{}^3 x-c_1{}^6+720 c_2{}^2 c_1{}^3+10368 c_2{}^4}} \\ y(x)\to \frac {-288 i \left (\sqrt {3}-i\right ) c_1 x^2+6 c_1{}^2 \sqrt [3]{10368 x^4+41472 c_2 x^3+720 c_1{}^3 x^2+62208 c_2{}^2 x^2+48 \sqrt {(x+c_2){}^2 \left (36 x^2+72 c_2 x-c_1{}^3+36 c_2{}^2\right ){}^3}+1440 c_2 c_1{}^3 x+41472 c_2{}^3 x-c_1{}^6+720 c_2{}^2 c_1{}^3+10368 c_2{}^4}+i \sqrt {3} \left (10368 x^4+41472 c_2 x^3+720 c_1{}^3 x^2+62208 c_2{}^2 x^2+48 \sqrt {(x+c_2){}^2 \left (36 x^2+72 c_2 x-c_1{}^3+36 c_2{}^2\right ){}^3}+1440 c_2 c_1{}^3 x+41472 c_2{}^3 x-c_1{}^6+720 c_2{}^2 c_1{}^3+10368 c_2{}^4\right ){}^{2/3}-\left (10368 x^4+41472 c_2 x^3+720 c_1{}^3 x^2+62208 c_2{}^2 x^2+48 \sqrt {(x+c_2){}^2 \left (36 x^2+72 c_2 x-c_1{}^3+36 c_2{}^2\right ){}^3}+1440 c_2 c_1{}^3 x+41472 c_2{}^3 x-c_1{}^6+720 c_2{}^2 c_1{}^3+10368 c_2{}^4\right ){}^{2/3}-576 i \left (\sqrt {3}-i\right ) c_2 c_1 x-i \sqrt {3} c_1{}^4-c_1{}^4-288 i \sqrt {3} c_2{}^2 c_1-288 c_2{}^2 c_1}{32 \sqrt [3]{10368 x^4+41472 c_2 x^3+720 c_1{}^3 x^2+62208 c_2{}^2 x^2+48 \sqrt {(x+c_2){}^2 \left (36 x^2+72 c_2 x-c_1{}^3+36 c_2{}^2\right ){}^3}+1440 c_2 c_1{}^3 x+41472 c_2{}^3 x-c_1{}^6+720 c_2{}^2 c_1{}^3+10368 c_2{}^4}} \\ y(x)\to \frac {288 i \left (\sqrt {3}+i\right ) c_1 x^2+6 c_1{}^2 \sqrt [3]{10368 x^4+41472 c_2 x^3+720 c_1{}^3 x^2+62208 c_2{}^2 x^2+48 \sqrt {(x+c_2){}^2 \left (36 x^2+72 c_2 x-c_1{}^3+36 c_2{}^2\right ){}^3}+1440 c_2 c_1{}^3 x+41472 c_2{}^3 x-c_1{}^6+720 c_2{}^2 c_1{}^3+10368 c_2{}^4}-i \sqrt {3} \left (10368 x^4+41472 c_2 x^3+720 c_1{}^3 x^2+62208 c_2{}^2 x^2+48 \sqrt {(x+c_2){}^2 \left (36 x^2+72 c_2 x-c_1{}^3+36 c_2{}^2\right ){}^3}+1440 c_2 c_1{}^3 x+41472 c_2{}^3 x-c_1{}^6+720 c_2{}^2 c_1{}^3+10368 c_2{}^4\right ){}^{2/3}-\left (10368 x^4+41472 c_2 x^3+720 c_1{}^3 x^2+62208 c_2{}^2 x^2+48 \sqrt {(x+c_2){}^2 \left (36 x^2+72 c_2 x-c_1{}^3+36 c_2{}^2\right ){}^3}+1440 c_2 c_1{}^3 x+41472 c_2{}^3 x-c_1{}^6+720 c_2{}^2 c_1{}^3+10368 c_2{}^4\right ){}^{2/3}+576 i \left (\sqrt {3}+i\right ) c_2 c_1 x+i \sqrt {3} c_1{}^4-c_1{}^4+288 i \sqrt {3} c_2{}^2 c_1-288 c_2{}^2 c_1}{32 \sqrt [3]{10368 x^4+41472 c_2 x^3+720 c_1{}^3 x^2+62208 c_2{}^2 x^2+48 \sqrt {(x+c_2){}^2 \left (36 x^2+72 c_2 x-c_1{}^3+36 c_2{}^2\right ){}^3}+1440 c_2 c_1{}^3 x+41472 c_2{}^3 x-c_1{}^6+720 c_2{}^2 c_1{}^3+10368 c_2{}^4}} \\ \end{align*}