Question

If y=2x + Ce^x is a solution of the differential equation dy/dx-y =2 (1-x) then find the particular solution satisfied by x=0, y=3

Answer 1

Answer on Question #86482 – Math – Differential Equations

Question

If $y = 2x + Ce^x$ is a solution of the differential equation $\frac{dy}{dx} - y = 2(1 - x)$ then find the particular solution satisfied by $x = 0, y = 3$.

Solution

Let's check that $y = 2x + Ce^x$ is a general solution of the differential equation

$\frac{dy}{dx} - y = 2(1 - x)$, by substitution it into the equation:

$$\frac{d}{dx} (2x + Ce^x) - (2x + Ce^x) = 2 + Ce^x - 2x - Ce^x = 2 - 2x = 2(1 - x)$$

Therefore $y = 2x + Ce^x$ is a general solution of the differential equation.

Then we find the particular solution by finding C when $x = 0, y = 3$, so

$$3 = 2 \cdot 0 + Ce^0 \Rightarrow C = 3$$

Thus, the particular solution is

$$y = 2x + 3e^x$$

**Answer**: The particular solution is $y = 2x + 3e^x$.

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