Answer to Question #343730 in Differential Equations for Vishu

Question #343730

Find the orthogonal trajectories of the following curves (where a is a parameter). r = a(1 + sin θ)

1
Expert's answer
2022-05-24T16:01:03-0400

Find the orthogonal trajectories of the following curves:

"r=a\\left(1+\\sin{\\theta}\\right)" ,

Solution:

"r\\prime=a\\cos{\\theta}" "\\Longrightarrow a=\\frac{r\\prime}{\\cos{\\theta}}"

then "r=\\frac{r\\prime}{\\cos{\\theta}}\\left(1+\\sin{\\theta}\\right)" .

For orthogonal trajectories in polar coordinates substitution is used: "r\\prime\\rightarrow-\\frac{r^2}{r\\prime}"

Then "r=-\\frac{r^2}{r\\prime}\\frac{1+\\sin{\\theta}}{\\cos{\\theta}}" ,

"\\frac{dr}{d\\theta}=-r\\frac{1+\\sin{\\theta}}{\\cos{\\theta}}" ,

"\\int\\frac{dr}{r}=-\\int{\\frac{1+\\sin{\\theta}}{\\cos{\\theta}}d\\theta}" ,

"\\ln{r}=-\\int{\\frac{1+\\sin{\\theta}}{\\cos{\\theta}}d\\theta}" ,

"\\int{\\frac{1+\\sin{\\theta}}{\\cos{\\theta}}d\\theta}=\\int{\\frac{\\left(1+\\sin{\\theta}\\right)\\cos{\\theta}}{{cos}^2\\theta}d\\theta}=\\int\\frac{\\left(1+\\sin{\\theta}\\right)d\\left(\\sin{x}\\right)}{1-{sin}^2\\theta}=\\int\\frac{d\\left(\\sin{x}\\right)}{1-\\sin{x}}"

"-\\int\\frac{d\\left(1-\\sin{x}\\right)}{1-\\sin{x}}\u00ac =-\\ln{\\left(1-\\sin{x}\\right)}+\\ln{C}" .

Answer:

"r=\u0421(1-sin\u03b8)"


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