Question

dy/dx -2y -4 =0 если y=2 при x=0 ​

Answer 1

$\frac{dy}{dx}-2y-4=0\\\\\frac{dy}{dx}=2y+4\\\\\int \frac{dy}{2y+4}=\int dx\\\\\frac{1}{2}\cdot ln|2y+4|=x+C\\\\y(0)=2:\; \; \frac{1}{2}\cdot ln|2\cdot 2+4|=0+C\; \; ,\; \; \frac{1}{2}\cdot ln8=C\; \; ,\; \; \frac{1}{2}\cdot 3\, ln2=C\\\\\frac{1}{2}\cdot ln|2y+4|=x+\frac{3\, ln2}{2}\; \; \Rightarrow \; \; ln|2y+4|=2x+3\, ln2\; \; \Rightarrow \\\\2y+4=e^{2x+3ln2}\; \; \Rightarrow \; \; y=\frac{1}{2}\, e^{2x+ln8}-2\; \; \Rightarrow \; \; y=\frac{1}{2}\, e^{2x}\cdot 8-2\\\\\underline {y=4\, e^{2x}-2}$