Use integration to find the particular solution of the differential equation which passes through the point

Question

Use integration to find the particular solution of the differential equation  which passes through the point

Quizplus · Accepted Answer

We can rewrite the given differential equation as $\frac{dy}{dx} + 2y = x^2$, with $y(0)=1$.
This is a linear first-order differential equation, which can be solved using integrating factor. 
The integrating factor is $e^{\int 2 dx} = e^{2x}$. 
Multiplying both sides of the differential equation by the integrating factor, we get: 
$$e^{2x}\frac{dy}{dx} + 2e^{2x}y = x^2e^{2x}$$
The left-hand side can now be rewritten using the chain rule: 
$$\frac{d}{dx}\left(e^{2x}y\right) = x^2e^{2x}$$
Integrating both sides with respect to x, we get: 
$$e^{2x}y = \frac{x^2}{2}e^{2x} + C$$
where C is a constant of integration. 
Using the initial condition $y(0)=1$, we can solve for C: 
$$e^{0}\cdot 1 = \frac{0^2}{2}\cdot e^{0} + C \Rightarrow C = 1$$
Substituting this value of C, we get: 
$$e^{2x}y = \frac{x^2}{2}e^{2x} + 1$$
Dividing both sides by $e^{2x}$, we get the particular solution: 
$$y = \frac{x^2}{2} + e^{-2x}$$
Substituting $y(0)=1$, we get: 
$$1 = \frac{0^2}{2} + e^{-2\cdot 0} \Rightarrow e^0 = 1$$
Therefore, the particular solution that passes through the point $(0,1)$ is $y=\frac{x^2}{2} + e^{-2x}$. 
The correct answer is (B). The given differential equation is:

$\frac{dy}{dx} = \frac{2x+3y+4}{3x+2y+6}$

To find the particular solution passing through the point (1,2), we need to use definite integration with the limits specified by the given point. Let's rearrange the given differential equation:

$(3x+2y+6)\frac{dy}{dx} = 2x+3y+4$

Now we can integrate both sides with respect to x:

$\int (3x+2y+6)\frac{dy}{dx} dx= \int(2x+3y+4)dx$

Integrating and simplifying:

$y(x) = \frac{3}{2}x + \frac{13}{2} + Ce^{-\frac{5}{3}x}$

Using the point (1,2), we get:

$2 = \frac{3}{2}(1) + \frac{13}{2} + Ce^{-\frac{5}{3}(1)}$

Solving for C:

$C = -\frac{2}{3}e^{\frac{5}{3}}$

So the particular solution passing through (1,2) is:

$y(x) = \frac{3}{2}x + \frac{13}{2} -\frac{2}{3}e^{-\frac{5}{3}x}$

Comparing with the given answer choices, we see that the correct option is (B).

Use Integration to Find the Particular Solution of the Differential