60.3.35 problem 1036

Internal problem ID [11045]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1036
Date solved : Monday, January 27, 2025 at 10:42:31 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+a y^{\prime }+b y-f \left (x \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 133

dsolve(diff(diff(y(x),x),x)+a*diff(y(x),x)+b*y(x)-f(x)=0,y(x), singsol=all)
 

\[ y = \frac {{\mathrm e}^{\frac {\left (-a +\sqrt {a^{2}-4 b}\right ) x}{2}} \left (\int f \,{\mathrm e}^{-\frac {\left (-a +\sqrt {a^{2}-4 b}\right ) x}{2}}d x \right )+{\mathrm e}^{\frac {\left (-a +\sqrt {a^{2}-4 b}\right ) x}{2}} c_{2} \sqrt {a^{2}-4 b}-{\mathrm e}^{-\frac {\left (a +\sqrt {a^{2}-4 b}\right ) x}{2}} \left (-\sqrt {a^{2}-4 b}\, c_{1} +\int f \,{\mathrm e}^{\frac {\left (a +\sqrt {a^{2}-4 b}\right ) x}{2}}d x \right )}{\sqrt {a^{2}-4 b}} \]

✓ Solution by Mathematica

Time used: 0.181 (sec). Leaf size: 152

DSolve[-f[x] + b*y[x] + a*D[y[x],x] + D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-\frac {1}{2} x \left (\sqrt {a^2-4 b}+a\right )} \left (\int _1^x-\frac {e^{\frac {1}{2} \left (a+\sqrt {a^2-4 b}\right ) K[1]} f(K[1])}{\sqrt {a^2-4 b}}dK[1]+e^{x \sqrt {a^2-4 b}} \int _1^x\frac {e^{\frac {1}{2} \left (a-\sqrt {a^2-4 b}\right ) K[2]} f(K[2])}{\sqrt {a^2-4 b}}dK[2]+c_2 e^{x \sqrt {a^2-4 b}}+c_1\right ) \]