12.9.34 problem 34

Internal problem ID [1790]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page 253
Problem number : 34
Date solved : Monday, January 27, 2025 at 05:34:53 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} y&=x \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&={\frac {5}{4}}\\ y^{\prime }\left (1\right )&={\frac {3}{2}} \end{align*}

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 11

dsolve([x^2*diff(diff(y(x),x),x)+2*x*diff(y(x),x)-2*y(x) = x^2, x, y(1) = 5/4, D(y)(1) = 3/2], singsol=all)
 

\[ y = x +\frac {1}{4} x^{2} \]

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 13

DSolve[x^2*D[y[x],{x,2}]+2*x*D[y[x],x]-2*y[x]==x^2,{y[1]==5/4,Derivative[1][y][1]==3/2},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{4} x (x+4) \]