(x^(3)+x^(2)+x+1)(dy)/(dx)=2x^(2)+x Y=1 When X=0

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Jul 03, 2024

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Solving the Differential Equation: (x^(3)+x^(2)+x+1)(dy)/(dx)=2x^(2)+x

Given the differential equation:

$(x^{3}+x^{2}+x+1)\frac{dy}{dx}=2x^{2}+x$

with the initial condition:

$y=1 \quad when \quad x=0$

Step 1: Simplify the Differential Equation

To start, we can simplify the left-hand side of the equation by expanding the product:

$x^{3}\frac{dy}{dx}+x^{2}\frac{dy}{dx}+x\frac{dy}{dx}+\frac{dy}{dx}=2x^{2}+x$

Step 2: Rearrange the Terms

Next, we can rearrange the terms to get:

$\frac{dy}{dx}(x^{3}+x^{2}+x+1)=2x^{2}+x$

Step 3: Divide Both Sides

Now, we can divide both sides of the equation by $(x^{3}+x^{2}+x+1)$ to get:

$\frac{dy}{dx}=\frac{2x^{2}+x}{x^{3}+x^{2}+x+1}$

Step 4: Integrate Both Sides

To solve for $y$, we can integrate both sides of the equation with respect to $x$:

$\int\frac{dy}{dx} dx=\int\frac{2x^{2}+x}{x^{3}+x^{2}+x+1} dx$

Step 5: Evaluate the Integral

Evaluating the integral on the right-hand side, we get:

$y=\ln|x^{3}+x^{2}+x+1|+C$

where $C$ is the constant of integration.

Step 6: Apply the Initial Condition

Using the initial condition $y=1$ when $x=0$, we can find the value of $C$:

$1=\ln|0^{3}+0^{2}+0+1|+C$

$C=1$

Step 7: Write the Final Solution

Finally, we can write the final solution as:

$y=\ln|x^{3}+x^{2}+x+1|+1$

Thus, we have solved the differential equation and found the general solution.