1.3 Examples for Frobenius series for first order ode

1.3.1 Example 1. \(xy^{\prime }+y=0,y\left ( 0\right ) =1\)
1.3.2 Example 2. \(xy^{\prime }+y=x\)
1.3.3 Example 3. \(xy^{\prime }+y=1\)
1.3.4 Example 4. \(xy^{\prime }+y=k\)
1.3.5 Example 5. \(xy^{\prime }+y=\sin x\)
1.3.6 Example 6. \(xy^{\prime }+y=x+x^{3}+2x^{4}\)
1.3.7 Example 7. \(xy^{\prime }+y=\frac {1}{x}\)
1.3.8 Example 8. \(xy^{\prime }+y=\frac {1}{x^{2}}\)
1.3.9 Example 9. \(xy^{\prime }+y=3+x\)
1.3.10 Example 10. \(xy^{\prime }+y=\frac {1}{x^{3}}\)
1.3.11 Example 11 \(y^{\prime }+2xy=\sqrt {x}\)
1.3.12 Example 12 \(y^{\prime }+2xy=\frac {1}{x}\)
1.3.13 Example 13 \(y^{\prime }=\frac {1}{x}\)
1.3.14 Example 14 \(y^{\prime }=\frac {1}{x^{2}}\)
1.3.15 Example 15 \(y^{\prime }+\frac {y}{x}=0\)
1.3.16 Example 16 \(\cos \left ( x\right ) y^{\prime }+\frac {1}{x}y=x\)
1.3.17 Example 17 \(xy^{\prime }+2xy=\sqrt {x}\)