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Magazine Problem 1
For each equation, list all the singular points in the finite plane. $$ \left(x^{2}+4\right) y^{\prime \prime}-6 x y^{\prime}+3 y=0 $$.
Problem 1
Unless requested otherwise, find the general solution valid near the origin. Always state the region of validity of the solution. Solve the equation \(y^{\prime \prime}+y=0\) both by series and by elementary methods and compare your answers.
Problem 2
For each equation, list all the singular points in the finite plane. $$ x(3-x) y^{\prime \prime}-(3-x) y^{\prime}+4 x y=0 $$.
Problem 2
Unless requested otherwise, find the general solution valid near the origin. Always state the region of validity of the solution. Solve the equation \(y^{\prime \prime}-9 y=0\) by series and by elementary methods.
Problem 4
Find the general solution valid near the origin. Always state the region of validity of the solution. $$ \left(1+4 x^{2}\right) y^{\prime \prime}-8 y=0 $$
Problem 4
For each equation, list all the singular points in the finite plane. $$ x(x-1)^{2} y^{\prime \prime}+3 x y^{\prime}+(x-1) y=0 $$.
Problem 5
Find the general solution valid near the origin. Always state the region of validity of the solution. $$ \left(1-4 x^{2}\right) y^{\prime \prime}+8 y=0 $$
Problem 5
For each equation, list all the singular points in the finite plane. $$ x^{2} y^{\prime \prime}+x y^{\prime}+\left(1-x^{2}\right) y=0 $$.
Problem 6
For each equation, list all the singular points in the finite plane. $$ x^{4} y^{\prime \prime}+y=0 $$.
Problem 6
Find the general solution valid near the origin. Always state the region of validity of the solution. $$ \left(1+x^{2}\right) y^{\prime \prime}-4 x y^{\prime}+6 y=0 $$
