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8. (a) Find $$\int (4y+3)^{-\frac{1}{2}} dy$$ (b) Given that $y = 1.5$ at $x = -2$, solve the differential equation $$\frac{dy}{dx} = \frac{\sqrt{(4y+3)}}{x^{2}}$$ giving your answer in the form $y = f(x)$. - Edexcel - A-Level Maths Pure - Question 4 - 2011 - Paper 5
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8. (a) Find $$\int (4y+3)^{-\frac{1}{2}} dy$$ (b) Given that $y = 1.5$ at $x = -2$, solve the differential equation $$\frac{dy}{dx} = \frac{\sqrt{(4y+3)... show full transcript
Worked Solution & Example Answer:8. (a) Find $$\int (4y+3)^{-\frac{1}{2}} dy$$ (b) Given that $y = 1.5$ at $x = -2$, solve the differential equation $$\frac{dy}{dx} = \frac{\sqrt{(4y+3)}}{x^{2}}$$ giving your answer in the form $y = f(x)$. - Edexcel - A-Level Maths Pure - Question 4 - 2011 - Paper 5
Step 1
Find $$\int (4y+3)^{-\frac{1}{2}} dy$$
Answer
To solve the integral, we make a substitution. Let thus, or . Substituting these into the integral, we have:
$$\int (4y+3)^{-\frac{1}{2}} dy = \int u^{-\frac{1}{2}} \frac{1}{4} du = \frac{1}{4} \cdot \int u^{-\frac{1}{2}} du$$
The integral of $u^{-\frac{1}{2}}$ is:
$$2u^{\frac{1}{2}} + C = 2(4y + 3)^{\frac{1}{2}} + C$$
Therefore, the solution to part (a) is:
$$\frac{1}{2}(4y+3)^{\frac{1}{2}} + C$$
Step 2
Given that $y = 1.5$ at $x = -2$, solve the differential equation $$\frac{dy}{dx} = \frac{\sqrt{(4y+3)}}{x^{2}}$$
Answer
Substituting the differential equation, we have:
$$\int \frac{1}{(4y + 3)^{\frac{1}{2}}} dy = \int \frac{1}{x^{2}} dx$$
The left side integrates to:
$$\frac{1}{2}(4y + 3)^{\frac{1}{2}} + C$$
The right side integrates to:
$$-\frac{1}{x} + C$$
Setting the two sides equal gives:
$$\frac{1}{2}(4y + 3) = -\frac{1}{x} + C$$
Now, substituting the initial conditions $(x = -2, y = 1.5)$:
$$\frac{1}{2}(4(1.5) + 3) = -\frac{1}{-2} + C$$
This leads to $C = \frac{1}{2}(6 + 3) - \frac{1}{2} = 4.5 - 0.5 = 4$$
Hence, the equation becomes:
$$\frac{1}{2}(4y + 3) = \frac{1}{2} - \frac{1}{x} + 4$$
Therefore,
$$y = \frac{1}{4} (2 - \frac{2}{x})^{2} - \frac{3}{4}$$
Simplifying gives:
$$y = \frac{(2 - \frac{2}{x})^{2}}{4} - \frac{3}{4}$$.