Chapter 12

For each of the following equations, one solution $u$ is given. Find the other solution by assuming a solution of the form $y=uv$. $x^2{y}^{\prime \prime }-3x{y}^{\prime }+4y=0;u=x^2$

For each of the following equations, one solution $u$ is given. Find the other solution by assuming a solution of the form $y=uv$. $x^3{y}^{\prime \prime }+x{y}^{\prime }-y=0;u=x$

For each of the following equations, one solution $u$ is given. Find the other solution by assuming a solution of the form $y=uv$. $(x^2+1)y^n-x{y}^{\prime }+y=0;u=x$

For each of the following equations, one solution $u$ is given. Find the other solution by assuming a solution of the form $y=uv$. $3x{y}^{\prime \prime }-2(3x-1)y^{\prime }+(3x-2)y=0;u=e^x$

For each of the following equations, one solution $u$ is given. Find the other solution by assuming a solution of the form $y=uv$. $x^2(2-x)y^{\prime \prime }+2x{y}^{\prime }-2y=0;u=x$

For each of the following equations, one solution $u$ is given. Find the other solution by assuming a solution of the form $y=uv$. $(x^2+1)y^{\prime \prime }-2x{y}^{\prime }+2y=0;u=x$