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Magazine Problem 73
Solve the differential equation using either the method of undetermined coefficients or the variation of parameters. $$y^{\prime \prime}+2 y^{\prime}=e^{3 x}$$
Problem 74
Solve the differential equation using either the method of undetermined coefficients or the variation of parameters. $$y^{\prime \prime}+6 y^{\prime}+9 y=e^{-x}$$
Problem 75
Solve the differential equation using either the method of undetermined coefficients or the variation of parameters. $$y^{\prime \prime}+2 y^{\prime}-8 y=6 e^{2 x}$$
Problem 76
Solve the differential equation using the method of variation of parameters. $$4 y^{\prime \prime}+y=2 \sin x$$
Problem 77
Solve the differential equation using the method of variation of parameters. $$y^{\prime \prime}-9 y=8 x$$
Problem 78
Solve the differential equation using the method of variation of parameters. $$y^{\prime \prime}+y=\sec x, \quad 0 < x < \pi / 2$$
Problem 79
Solve the differential equation using the method of variation of parameters. $$y^{\prime \prime}+4 y=3 \csc 2 x, \quad 0 < x < \pi / 2$$
Problem 80
Find the unique solution satisfying the differential equation and the initial conditions given, where \(y_{p}(x)\) is the particular solution. \(y^{\prime \prime}-2 y^{\prime}+y=12 e^{x}\), \(\mathrm{y}_{p}(x)=6 x^{2} e^{x}\), \(y(0)=-1, y^{\prime}(0)=0\)
Problem 81
Find the unique solution satisfying the differential equation and the initial conditions given, where \(y_{p}(x)\) is the particular solution. \(y^{\prime \prime}-7 y^{\prime}=4 x e^{7 x}\), \(y_{p}(x)=\frac{2}{7} x^{2} e^{7 x}-\frac{4}{49} x e^{7 x}\), \(y(0)=-1,= y^{\prime}(0)=0\)
Problem 82
Find the unique solution satisfying the differential equation and the initial conditions given, where \(y_{p}(x)\) is the particular solution. \(y^{\prime \prime}+y=\cos x-4 \sin x\), \(y_{p}(x)=2 x \cos x+\frac{1}{2} x \sin x, \quad y(0)=8, \quad y^{\prime}(0)=-4\)
