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Magazine Problem 18
In each of the following eliminate the arbitrary constants. $$ y=c_{1} e^{x}+c_{2} x e^{x} $$
Problem 19
In each of the following eliminate the arbitrary constants. $$ y=A e^{2 x}+B x e^{2 x} $$
Problem 20
In each of the following eliminate the arbitrary constants. $$ y=c_{1} e^{2 x} \cos 3 x+c_{2} e^{2 x} \sin 3 x $$
Problem 21
In each exercise, obtain the differential equation of the family of plane curves described and sketch several representative members of the family. Parabolas with axis parallel to the \(x\) -axis.
Problem 21
In each of the following eliminate the arbitrary constants. $$ y=c_{1} e^{a x} \cos b x+c_{2} e^{a x} \sin b x ; a \text { and } b \text { are parameters. } $$
Problem 22
In each exercise, obtain the differential equation of the family of plane curves described and sketch several representative members of the family. Central conics with center at the origin and vertices on the coordinate axes.
Problem 22
In each of the following eliminate the arbitrary constants. $$ y=c_{1} x+c_{2} e^{x} $$
Problem 23
In each exercise, obtain the differential equation of the family of plane curves described and sketch several representative members of the family. The confocal central conics $$ \frac{x^{2}}{a^{2}+\lambda}+\frac{y^{2}}{b^{2}+\lambda}=1 $$ with \(a\) and \(b\) held fixed.
Problem 24
In each of the following eliminate the arbitrary constants. $$ y=x^{2}+c_{1} x+c_{2} e^{-x} $$
Problem 24
In each exercise, obtain the differential equation of the family of plane curves described and sketch several representative members of the family. The cubics \(c y^{2}=x^{2}(x-a)\) with \(a\) held fixed.
