Find all solutions of the given differential equations and then find the particular solutions for which a point (x,y) is given:
dy/dx = √x + 3 ; (x,y) =(-1,3)
First we solve the integral by separating the terms:
"\\frac{dy}{dx}=\\sqrt{x}+3 \n\\\\ \\implies \\intop dy=\\int(\\sqrt{x}+3)dx\n\\\\y=\\int\\ x^{1\/2}dx+3\\int dx=\\frac{x^{3\/2}}{3\/2}+3x+C\n\\\\y=\\frac{2x^{3\/2}}{3}+3x+C"
Then, we substitute the coordinates (x,y) to find C for the particular solution:
"3=\\frac{2(-1)^{3\/2}}{3}+3(-1)+C \n\\\\ \\implies 3=-\\frac{2i}{3}-3+C\n\\\\ \\to C=6+\\frac{2i}{3}"
In conclusion:
"\\text{General solution: }\n\\\\ y=\\frac{2x^{3\/2}}{3}+3x+C\n\\\\ \\text{Particular solution: }\n\\\\ y=\\frac{2x^{3\/2}}{3}+3x+6+\\frac{2i}{3}"
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