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Exercise2 x y d x = − x 2 d y 2xydx=-x^2dy 2 x y d x = − x 2 d y
Step-by-step Solution
1
Divide fractions 1 − x 2 x \frac{1}{\frac{-x^2}{x}} x − x 2 1 a ÷ b c = a 1 ÷ b c = a 1 × c b = a ⋅ c b a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b} a ÷ c b = 1 a ÷ c b = 1 a × b c = b a ⋅ c
1 2 y = x − x 2 \frac{1}{2y}=\frac{x}{-x^2} 2 y 1 = − x 2 x
2
Integrate both sides of the differential equation, the left side with respect to y y y x x x
∫ 1 2 y d y = ∫ 1 − x d x \int\frac{1}{2y}dy=\int\frac{1}{-x}dx ∫ 2 y 1 d y = ∫ − x 1 d x
Intermediate steps
3
Solve the integral ∫ 1 2 y d y \int\frac{1}{2y}dy ∫ 2 y 1 d y
1 2 ln ∣ y ∣ = ∫ 1 − x d x \frac{1}{2}\ln\left|y\right|=\int\frac{1}{-x}dx 2 1 ln ∣ y ∣ = ∫ − x 1 d x
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Intermediate steps
4
Solve the integral ∫ 1 − x d x \int\frac{1}{-x}dx ∫ − x 1 d x
1 2 ln ∣ y ∣ = − ln ∣ x ∣ + C 0 \frac{1}{2}\ln\left|y\right|=-\ln\left|x\right|+C_0 2 1 ln ∣ y ∣ = − ln ∣ x ∣ + C 0
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Intermediate steps
5
Find the explicit solution to the differential equation. We need to isolate the variable y y y
y = C 2 x 2 y=\frac{C_2}{x^{2}} y = x 2 C 2
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Final answer to the exercise y = C 2 x 2 y=\frac{C_2}{x^{2}} y = x 2 C 2