PIL import Image import pytesseract

Load the uploaded image

image_path = '/mnt/data/1736936480049rm4473n1.jpg' image = Image.open(image_path)

Use OCR to extract the mathematical content

extracted_text = pytesseract.image_to_string(image, config='--psm 6')

extracted_textThe equation extracted from the image is approximately:

$2 \frac{dy}{dt} + \frac{1}{t} y = 1 + \sqrt{t} e^t \quad (\text{for } t > 0).$

Let us solve this step by step. This is a first-order linear differential equation.

Step 1: Write the equation in standard form

Divide the entire equation by 2 to make the coefficient of $\frac{dy}{dt}$ equal to 1: $\frac{dy}{dt} + \frac{1}{2t} y = \frac{1}{2} + \frac{\sqrt{t}}{2} e^t.$

Step 2: Find the integrating factor

The integrating factor is given by: $\mu(t) = e^{\int \frac{1}{2t} dt} = e^{\frac{1}{2} \ln t} = t^{1/2}.$

Step 3: Multiply through by the integrating factor

Multiply the entire equation by $t^{1/2}$: $t^{1/2} \frac{dy}{dt} + \frac{t^{1/2}}{2t} y = t^{1/2} \left(\frac{1}{2} + \frac{\sqrt{t}}{2} e^t \right).$

This simplifies to: $\frac{d}{dt} \left( t^{1/2} y \right) = \frac{t^{1/2}}{2} + \frac{t}{2} e^t.$

Step 4: Integrate both sides

Integrate both sides to find $y$: $t^{1/2} y = \int \frac{t^{1/2}}{2} dt + \int \frac{t}{2} e^t dt.$

First integral:

$\int \frac{t^{1/2}}{2} dt = \frac{1}{2} \cdot \frac{2}{3} t^{3/2} = \frac{1}{3} t^{3/2}.$

Second integral:

For $\int \frac{t}{2} e^t dt$, we use integration by parts: Let $u = t$, $dv = \frac{1}{2} e^t dt$, so $du = dt$ and $v = \frac{1}{2} e^t$.

Then: $\int \frac{t}{2} e^t dt = \frac{1}{2} t e^t - \int \frac{1}{2} e^t dt = \frac{1}{2} t e^t - \frac{1}{2} e^t = \frac{1}{2} e^t (t - 1).$

Step 5: Combine the results

Now, substitute back into the equation for $t^{1/2} y$: $t^{1/2} y = \frac{1}{3} t^{3/2} + \frac{1}{2} e^t (t - 1) + C,$ where $C$ is the constant of integration.

Step 6: Solve for $y$

Divide through by $t^{1/2}$ to isolate $y$: $y = \frac{1}{3} t + \frac{1}{2} t^{1/2} e^t (t - 1) + \frac{C}{t^{1/2}}.$

This is the solution.

Would you like more details about any step?

Related Questions

1. What are some other methods to solve linear differential equations?
2. How do you handle cases where $t = 0$ in the context of the integrating factor?
3. What is the role of the integrating factor in simplifying these equations?
4. How does integration by parts work in a general sense?
5. Can this method be extended to second-order differential equations?

Tip

Always check the form of the differential equation before proceeding—deciding if it is linear, separable, or exact can simplify the solution process.