Solution of Euler-Cauchy equation, simultaneous differential equations by operator method

Correct Answer: a) ax²y'' + bxy' + cy = 0

Explanation:

1. The **Euler-Cauchy equation** is a second-order linear differential equation that has the general form:

\( ax^2y'' + bxy' + cy = 0 \)

2. This form is distinct because it involves variable coefficients, specifically powers of \( x \) multiplying the derivatives.

3. It is often used in applications where the behavior of solutions changes based on powers of the independent variable.

4. By applying the substitution \( x = e^t \) or \( t = \ln x \), the Euler-Cauchy equation can be transformed into a constant coefficient differential equation.

5. Other forms like \( ay'' + by' + cy = 0 \) are characteristic of constant coefficient equations, not Euler-Cauchy equations.

Hence, the correct answer is: Option (a).

Correct Answer: b) y = x^r

Explanation:

1. The standard form of an **Euler-Cauchy equation** is:

\( ax^2y'' + bxy' + cy = 0 \)

2. To solve this, we typically use the substitution:

\( y = x^r \)

3. This choice is effective because the characteristic form of the equation leads to an algebraic equation called the **auxiliary (characteristic) equation**:

\( ar(r-1) + br + c = 0 \)

4. Solving this characteristic equation gives the values of \( r \), which are used to form the general solution.

5. Other substitutions like \( y = e^{rx} \) or \( y = \sin(rx) \) are generally used for constant coefficient differential equations, not for Euler-Cauchy equations.

Hence, the correct answer is: Option (b).

Correct Answer: b) r² + 3r + 2 = 0

Explanation:

1. The standard form of an **Euler-Cauchy equation** is:

\( x^2y'' + bxy' + cy = 0 \)

2. To solve it, we use the substitution:

\( y = x^r \)

3. Then the derivatives become:

\( y' = rx^{r-1}, \quad y'' = r(r-1)x^{r-2} \)

4. Substituting these into the given equation:

\( x^2 \cdot r(r-1)x^{r-2} + 4x \cdot rx^{r-1} + 2x^r = 0 \)

5. Simplify:

\( r(r-1)x^r + 4rx^r + 2x^r = 0 \)

6. Factor out \( x^r \) (assuming \( x \neq 0 \)):

\( x^r \left( r^2 + 4r + 2 \right) = 0 \)

7. The characteristic equation is:

\( r^2 + 4r + 2 = 0 \)

Hence, the correct answer is: Option (a).

Correct Answer: a) x^rlnx

Explanation:

1. The general form of an **Euler-Cauchy equation** is:

\( ax^2y'' + bxy' + cy = 0 \)

2. Using the substitution \( y = x^r \), we derive the characteristic equation:

\( ar(r-1) + br + c = 0 \)

3. If the characteristic equation has **repeated roots** (say \( r_1 = r_2 = r \)), then the first solution is:

\( y_1 = x^r \)

4. In case of repeated roots, the second solution is given by a special form:

\( y_2 = x^r \ln x \)

5. This form is necessary because applying the usual \( x^r \) form for both roots would not produce a linearly independent solution.

Hence, the correct answer is: Option (a).

Correct Answer: a) x^α(c₁cos(βlnx) + c₂sin(βlnx))

Explanation:

1. The general form of an **Euler-Cauchy equation** is:

\( ax^2y'' + bxy' + cy = 0 \)

2. Substituting \( y = x^r \) results in the characteristic equation:

\( ar(r-1) + br + c = 0 \)

3. For complex roots \( \alpha \pm i\beta \), the general solution takes the form:

\( y = x^\alpha \left(c_1 \cos(\beta \ln x) + c_2 \sin(\beta \ln x)\right) \)

4. This form arises because \( \ln x \) naturally appears when solving with complex exponents in the Euler-Cauchy case.

Hence, the correct answer is: Option (a).

Correct Answer: b) An algebraic operator

Explanation:

1. In the operator method, \( D \) represents the differential operator:

\( D = \frac{d}{dt} \)

2. This allows the differential equations to be treated like algebraic equations, where the operator \( D \) behaves similarly to a variable in polynomial form.

3. By applying algebraic manipulations, we can solve for the functions using characteristic equations or inverse operators.

4. Therefore, \( D \) is treated as an **algebraic operator** in the operator method.

Hence, the correct answer is: Option (b).

Correct Answer: c) Operate on equations to eliminate one variable

Explanation:

1. The goal is to eliminate one of the variables to reduce the system to a single differential equation.

2. The most effective approach is to apply **operator manipulations** to combine the two equations.

3. By adding or subtracting, or applying other algebraic manipulations, one variable can be eliminated, simplifying the problem to a single differential equation in one variable.

4. This process is a fundamental application of the operator method.

Hence, the correct answer is: Option (c).

Correct Answer: b) x = c₁cost + c₂sint, y = -c₁sint + c₂cost

Explanation:

1. The given system of differential equations can be written as:

\(\frac{dx}{dt} = y \quad \text{and} \quad \frac{dy}{dt} = -x\)

2. Differentiating the first equation: \(\frac{d^2x}{dt^2} = \frac{dy}{dt} = -x\)

3. This gives the second-order differential equation: \(\frac{d^2x}{dt^2} + x = 0\)

4. The characteristic equation is: \(r^2 + 1 = 0 \implies r = \pm i\)

5. Therefore, the general solution is of the form:

\(x = c_1\cos t + c_2\sin t\)

\(y = -c_1\sin t + c_2\cos t\)

6. This corresponds to the correct answer.

Hence, the correct answer is: Option (b).

Correct Answer: b) Cramer's rule

Explanation:

1. The determinant method is a systematic way of solving simultaneous linear differential equations using operators.

2. This method is directly analogous to Cramer’s rule, which is a well-known algebraic technique for solving linear equations using determinants.

3. In the context of differential equations, determinants are used to eliminate variables and derive particular or general solutions.

4. Other options like integration by parts, partial fractions, and variation of parameters are not directly related to the determinant method.

5. Therefore, the correct answer is:

Option (b).

Correct Answer: c) Both a and b

Explanation:

1. To solve non-homogeneous simultaneous equations, the standard approach involves two main steps:

- First, solve the corresponding **homogeneous system** to obtain the complementary function (general solution of the homogeneous part).

- Then, find the **particular solution** using methods like the method of undetermined coefficients or variation of parameters.

2. The final solution is the sum of the complementary function and the particular solution.

3. Matrix inversion is typically not the main approach for directly solving non-homogeneous differential equations.

4. Therefore, the correct answer is:

Option (c).