83.43.13 problem Ex 14 page 15

Internal problem ID [19521]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Book Solved Excercises. Chapter II. Equations of first order and first degree
Problem number : Ex 14 page 15
Date solved : Tuesday, January 28, 2025 at 01:48:16 PM
CAS classification : [_Bernoulli]

\begin{align*} y^{\prime }+p \left (x \right ) y&=q \left (x \right ) y^{n} \end{align*}

✓ Solution by Maple

Time used: 0.059 (sec). Leaf size: 55

dsolve(diff(y(x),x)+p(x)*y(x)=q(x)*y(x)^n,y(x), singsol=all)
 

\[ y \left (x \right ) = {\mathrm e}^{-\int p \left (x \right )d x} {\left (-n \left (\int q \left (x \right ) {\mathrm e}^{-\left (\int p \left (x \right )d x \right ) \left (n -1\right )}d x \right )+c_{1} +\int q \left (x \right ) {\mathrm e}^{-\left (\int p \left (x \right )d x \right ) \left (n -1\right )}d x \right )}^{-\frac {1}{n -1}} \]

✓ Solution by Mathematica

Time used: 12.507 (sec). Leaf size: 137

DSolve[D[y[x],x]+p[x]*y[x]==q[x]*y[x]^n,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \left (\exp \left (-\left ((n-1) \int _1^x-p(K[1])dK[1]\right )\right ) \left (-(n-1) \int _1^x\exp \left ((n-1) \int _1^{K[2]}-p(K[1])dK[1]\right ) q(K[2])dK[2]+c_1\right )\right ){}^{\frac {1}{1-n}} \\ y(x)\to \left ((n-1) \left (-\exp \left (-\left ((n-1) \int _1^x-p(K[1])dK[1]\right )\right )\right ) \int _1^x\exp \left ((n-1) \int _1^{K[2]}-p(K[1])dK[1]\right ) q(K[2])dK[2]\right ){}^{\frac {1}{1-n}} \\ \end{align*}