Differential Equations

To solve the problem, we need to evaluate the integral and find for the form .

1. First Integration by Parts:
Let , .
Then , .
.

2. Second Integration by Parts:
For , let , .
Then , .
.

3. Third Integration by Parts:
For , let , .
Then , .
.

4. Combine Results:
Substitute back:
.
Then:
.
Simplify:
.

5. Identify Coefficients:
Compare with :
, , , .

6. Compute the Final Expression:
Calculate :
.
.

Final Answer:
The value of is .

To solve the problem, we need to evaluate the integral .

1. Decompose the Integrand:
Write .
Multiply through by :
.
Equate coefficients:
, .

2. Solve for Coefficients:
Subtract from :
, so .
Then .

3. Rewrite the Integral:
Substitute back:
.

4. Evaluate the Integrals:
The first integral is:
.
The second integral is:
.
Thus:
.

Final Answer:
The integral is .

To solve the problem, we need to evaluate the integral .

1. Rewrite the Integrand:
Rewrite the denominator:
.
Thus:
.

2. Substitution:
Let , so .
Solve for :
, so , and .
Thus, .

3. Compute the Differential:
Differentiate :
.

4. Express in Terms of :
.

5. Substitute and Simplify:
Substitute into the integral:
.
Integrate:
.

Final Answer:
The integral is .

To solve the problem, we need to find the order and degree of the differential equation for the family of curves , where is a parameter, and compute .

1. Differentiate the Equation:
Given , differentiate with respect to :
.
Solve for :
.

2. Eliminate the Parameter:
Substitute into :
.
Simplify:
.
Divide by (assuming ):
.
Expand:
.

3. Determine Order and Degree:
The differential equation is .
The highest-order derivative is , so the order is .
The highest power of is , so the degree is .

4. Compute :
Calculate:
.

Final Answer:
The value of is .

To solve the problem, we need to evaluate the given options about the direction cosines of two lines and identify the correct statement.

1. Understand Direction Cosines Properties:
Let and be the direction cosines of two lines.
Two lines are parallel if their direction cosines are proportional: .
Two lines are perpendicular if the dot product of their direction cosines is zero: .
The direction ratios of the angle bisectors between the lines are .

2. Evaluate the Given Options:
Option 1 states the lines are parallel when .
This is incorrect, as indicates perpendicularity, not parallelism.
Option 2 states the lines are perpendicular when .
This is incorrect, as indicates parallelism, not perpendicularity.
Option 3 states the direction ratios of the angle bisectors are .
This is correct, as it matches the standard formula for the direction ratios of angle bisectors.

3. Conclusion:
Options 1 and 2 are incorrect, but option 3 is correct.

Final Answer:
The direction ratios of the angle bisectors are .

To solve the problem, we need to find for functions derived from the implicit differentiation of the equation .

1. Differentiate the Equation:
Differentiate with respect to :
.
Rearrange terms:
.
Thus, .

2. Identify and :
We want the differential form , so .
Comparing with , we get:
and .

3. Compute :
Evaluate .
Evaluate .
Thus, .

Final Answer:
The value of is .

To solve the problem, we need to find the constant in the integral .

1. Set Up the Substitution:
Let .
Then .
Thus, .

2. Rewrite the Integral:
Substitute into the integral:
.

3. Evaluate the Integral:
Integrate:
.
Substitute back :
.

4. Determine :
The integral is given as .
Comparing with , we find .

Final Answer:
The value of is .

Step 1: Rewrite the Equation in a Simpler Form

We begin by rewriting the equation in a more convenient form. Notice that the left-hand side resembles the derivative of a product. Specifically:

(x² + 2) dy + 2xy dx = d/dx (x² + 2)y

Thus, the equation becomes:

d/dx (x² + 2)y = e^x (x² + 2) dx

Step 2: Integrate Both Sides

Now that we have the equation in terms of a derivative, we can integrate both sides. Let’s do the integration step by step.

• Left-hand side: The integral of the derivative is just (x² + 2)y.
• Right-hand side: We need to integrate e^x (x² + 2) with respect to x. Let’s break it down into two simpler integrals:

∫ e^x (x² + 2) dx = ∫ e^x x² dx + ∫ 2 e^x dx

Step 3: Integrate Each Term on the Right-Hand Side

1. First, solve ∫ e^x x² dx using integration by parts:

• Let u = x² and dv = e^x dx.
• Then, du = 2x dx and v = e^x.
• Apply the integration by parts formula: ∫ u dv = uv - ∫ v du

2. Next, solve ∫ 2x e^x dx using integration by parts:

• Let u = 2x and dv = e^x dx.
• Then, du = 2 dx and v = e^x.
• Apply the integration by parts formula again:

Now, putting everything together:

∫ e^x x² dx = x² e^x - (2x e^x - 2 e^x) = x² e^x - 2x e^x + 2 e^x

Step 4: Combine Everything

Now combine all the terms from the integration:

∫ e^x (x² + 2) dx = e^x (x² - 2x + 4) + C

Step 5: Write the Final Solution

Now, we can write the general solution. Recall the equation we had earlier:

(x² + 2)y = e^x (x² - 2x + 4) + C

Conclusion

We have solved the given differential equation and arrived at the general solution. By following these steps, you can solve similar first-order linear differential equations and understand the techniques involved in integrating complex expressions.

To solve the problem, we need to find the general solution of the homogeneous differential equation and identify the coefficient of in the expression for .

1. Verify Homogeneity:
The equation is homogeneous since .
Let , so .

2. Substitute and Simplify:
Substitute into the equation:
.
Solve for :
.
Thus:
.

3. Integrate Both Sides:
Left-hand side:
.
Right-hand side:
.
Equate:
.

4. Substitute Back:
Since :
.
Simplify:
.
Thus:
.

5. Simplify the Solution:
Multiply by 2:
.
Rewrite:
.
The coefficient of is .

Final Answer:
The coefficient of is .