Chapter 12

$x{y}^{\prime \prime }+2y^{\prime }+4y=0$

$(d/dx)J_0(x)=-J_1(x)$

$3x{y}^{\prime \prime }+2y^{\prime }+12y=0$

Solve the following differential equations by power series and also by an elementary method. Verify that the series solution is the power series expansion of your other solution. $$y^{\prime \prime }+y=4\sin 3x$$

Evaluate ${\int }_{-1}^1,P_0(x)P_2(x)dx$ to show that these functions are orthogonal on $(-1,1)$.

Solve the following differential equations by power series and also by an elementary method. Verify that the series solution is the power series expansion of your other solution. $$y^{\prime \prime }-2y^{\prime }+y=0$$

Solve the following differential equations by power series and also by an elementary method. Verify that the series solution is the power series expansion of your other solution. $$x{y}^{\prime }-y=x^2$$

$3x{y}^{\prime \prime }+(3x+1)y^{\prime }+y=0$

Solve the following differential equations by power series and also by an elementary method. Verify that the series solution is the power series expansion of your other solution. $$y^{\prime }=3x^{2}y$$

$\underset{x\to 0}{lim}{J}_1(x)/x=\frac{1}{2}$