9.31 problem 31

Internal problem ID [1137]

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.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]

Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }-3 y^{\prime } x +4 y-4 x^{4}=0} \end {gather*} Given that one solution of the ode is \begin {align*} y_1 &= 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: 0.023 (sec). Leaf size: 21

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],y(x), singsol=all)
 

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

Solution by Mathematica

Time used: 0.01 (sec). Leaf size: 32

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

\begin{align*} y(x)\to x^2 (2 \log (x) (\log (x)-2 i \pi -11)-2 \pi (\pi -11 i)+7) \\ \end{align*}