12.9.31 problem 31

Internal problem ID [1787]
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 : 31
Date solved : Monday, January 27, 2025 at 05:34:51 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y&=4 x^{4} \end{align*}

Using reduction of order method given that one solution is

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

With initial conditions

\begin{align*} y \left (-1\right )&=7\\ y^{\prime }\left (-1\right )&=-8 \end{align*}

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 20

dsolve([x^2*diff(diff(y(x),x),x)-3*x*diff(y(x),x)+4*y(x) = 4*x^4, x^2, y(-1) = 7, D(y)(-1) = -8], singsol=all)
 

\[ y = x^{2} \left (8 i \pi +x^{2}-8 \ln \left (x \right )+6\right ) \]

✓ Solution by Mathematica

Time used: 0.022 (sec). Leaf size: 37

DSolve[x^2*D[y[x],{x,2}]-3*x*D[y[x],x]+4*y[x]==4*x^2,{y[-1]==7,Derivative[1][y][-1]==8},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to x^2 \left (2 \log ^2(x)+(-22-4 i \pi ) \log (x)-2 \pi ^2+22 i \pi +7\right ) \]