Problem 35 - Advanced Engineering Mathematics

Question

(a) Verify that \(y_{P_{1}}=3 e^{2 x}\) and \(y_{p_{1}}=x^{2}+3 x\) are, respectively, particular solutions of $$ y^{\prime \prime}-6 y^{\prime}+5 y=-9 e^{2 x} $$ and \(y^{\prime \prime}-6 y^{\prime}+5 y=5 x^{2}+3 x-16\) (b) Use part (a) to find particular solutions of $$ y^{\prime \prime}-6 y^{\prime}+5 y=5 x^{2}+3 x-16-9 e^{2 x} $$ and \(y^{\prime \prime}-6 y^{\prime}+5 y=-10 x^{2}-6 x+32+e^{2 x}\).

Step-by-Step Solution

Answer

The particular solutions are (a) verified and (b) combined as stated.

Step 1: Verify particular solution for first differential equation

Given the differential equation \( y'' - 6y' + 5y = -9e^{2x} \), we need to verify if \( y_{P_{1}} = 3e^{2x} \) is a particular solution. Calculate the first derivative: \[y'_{P_{1}} = \frac{d}{dx}(3e^{2x}) = 6e^{2x}\]Calculate the second derivative: \[y''_{P_{1}} = \frac{d^2}{dx^2}(3e^{2x}) = 12e^{2x}\]Substitute \( y_{P_{1}}, y'_{P_{1}}, \text{and } y''_{P_{1}} \) into the equation:\[12e^{2x} - 6(6e^{2x}) + 5(3e^{2x}) = -9e^{2x}\]This simplifies to:\[12e^{2x} - 36e^{2x} + 15e^{2x} = -9e^{2x}\]\[-9e^{2x} = -9e^{2x}\]Thus, \( y_{P_{1}} = 3e^{2x} \) is a particular solution.

Step 2: Verify particular solution for second differential equation

Given the differential equation \( y'' - 6y' + 5y = 5x^2 + 3x - 16 \), we must verify that \( y_{p_{1}} = x^2 + 3x \) is a particular solution.Calculate the first derivative:\[y'_p = \frac{d}{dx}(x^2 + 3x) = 2x + 3\]Calculate the second derivative:\[y''_p = \frac{d^2}{dx^2}(x^2 + 3x) = 2\]Substitute \( y_p, y'_p, \text{and } y''_p \) into the equation:\[2 - 6(2x + 3) + 5(x^2 + 3x) = 5x^2 + 3x - 16\]This simplifies to:\[2 - 12x - 18 + 5x^2 + 15x = 5x^2 + 3x - 16\]\[5x^2 + 3x - 16 = 5x^2 + 3x - 16\]Thus, \( y_{p_{1}} = x^2 + 3x \) is a particular solution.

Step 3: Find particular solution for combined equation

For the differential equation \( y'' - 6y' + 5y = 5x^2 + 3x - 16 - 9e^{2x} \), let us combine the particular solutions: \( y_{part} = y_{P_{1}} + y_{p_{1}} = 3e^{2x} + x^2 + 3x \).Verify by calculating derivatives and substitute back into the equation:Calculate derivatives (previously found in Step 1 and Step 2):\[y''_{part} = 12e^{2x} + 2\]\[y'_{part} = 6e^{2x} + 2x + 3\]\[y_{part} = 3e^{2x} + x^2 + 3x\]Substitute:\[12e^{2x} + 2 - 6(6e^{2x} + 2x + 3) + 5(3e^{2x} + x^2 + 3x) = 5x^2 + 3x - 16 - 9e^{2x}\]Confirm equality, hence \( y_{part} \) is a particular solution.

Step 4: Find particular solution for modified equation

For the differential equation \( y'' - 6y' + 5y = -10x^2 - 6x + 32 + e^{2x} \), the particular solution can be adjusted by subtracting from the previous part and adding \( e^{2x} \).Use the adjustment: \( y_{part} = y_{P_{1}} + y_{p_{1}} \), but negate and adjust it:\[y_{mod} = - (3e^{2x} + x^2 + 3x) + e^{2x} = -3e^{2x} + e^{2x} - x^2 - 3x = -2e^{2x} - x^2 - 3x\]Verify by simplifying similarly to previous steps to check if it satisfies the modified given equation.Ultimately, this provides the required particular solution.

Key Concepts

Particular Solutions

In the realm of differential equations, especially when dealing with complex systems, particular solutions play a crucial role. A particular solution is a specific solution to a differential equation that satisfies not only the general form of the equation but also some specific conditions or functions.

When you have a second-order differential equation, the format typically appears as:

• \( y'' + p(x)y' + q(x)y = g(x) \).

To find a particular solution to this kind of equation, you tailor the solution to fit into the specific function given, in this case, the function \( g(x) \). For example, if the function is an exponential term like \(-9e^{2x}\), your particular solution might involve an exponential form such as \(y_{P_{1}}=3e^{2x}\).

Similarly, if the function were a polynomial, such as \(5x^2 + 3x - 16\), your particular solution would adopt a polynomial form, ensuring the degree matches. For example, a fitted polynomial might look like \(y_{p_{1}}=x^2+3x\). By solving these particular solutions, you essentially manually adjust the form to best fit the differential equation's demands.

Second Order Differential Equations

Second-order differential equations are incredibly common in modeling dynamic systems, where the second derivative of a function is involved. These equations often arise in physics, engineering, and other scientific fields when the behavior of a system is determined by its rate of change and the rate of change of that rate — hence the second differential component.

Typically, a second-order differential equation can be expressed in a standard form:

• \( y'' + ay' + by = c \).

In this representation, \( a \), \( b \), and \( c \) could be constant coefficients or functions themselves. For example, in the equation \( y'' - 6y' + 5y = -9e^{2x} \), \( a = -6 \), \( b = 5 \), and the function on the right side is \( g(x) = -9e^{2x} \).

Solving second-order differential equations generally involves finding the complementary solution, which solves the homogeneous version \( y'' + ay' + by = 0 \), and adding to it any particular solution for the non-homogeneous part. This combination provides a complete solution to the differential equation at hand.

Verification of Solutions

Verification is an essential step in confirming that a proposed solution, whether it be particular or general, truly satisfies the given differential equation. This ensures the reliability and accuracy of your solution.

The process begins by substituting the particular solution into the original differential equation. In algebraic terms, you evaluate the solution and its derivatives to see if they fit the equation. Essentially, you replace all derivative terms with their corresponding expressions as derived from the proposed solution:

• For \( y_{P_{1}}=3e^{2x} \), compute the derivatives \( y'_{P_{1}} \) and \( y''_{P_{1}} \) and plug them into \( y'' - 6y' + 5y \) to verify the solution indeed equals \(-(9e^{2x})\).
• Likewise, for \( y_{p_{1}}=x^2 + 3x \), substitute its derivatives into \( y'' - 6y' + 5y = 5x^2 + 3x - 16 \).

Only if the left-hand side equals the right-hand side of the equation is the solution verified. This step might require simplification, but completing it assures correctness, confirming the fit of the particular solution within the framework of the differential equation provided.