Question

Assume that \(u(t)\) and \(v(t)\) are, respectively, solutions of the differential equations $$ u^{\prime \prime}+p(t) u^{\prime}+q(t) u=g_{1}(t) \quad \text { and } \quad v^{\prime \prime}+p(t) v^{\prime}+q(t) v=g_{2}(t), $$ where \(p(t), q(t), g_{1}(t)\), and \(g_{2}(t)\) are continuous on the \(t\)-interval of interest. Let \(a_{1}\) and \(a_{2}\) be any two constants. Show that the function \(y_{p}(t)=a_{1} u(t)+a_{2} v(t)\) is a particular solution of the differential equation $$ y^{\prime \prime}+p(t) y^{\prime}+q(t) y=a_{1} g_{1}(t)+a_{2} g_{2}(t) $$

Short answer

Question: Show that if u(t) and v(t) are solutions to the differential equations \(u''(t) + p(t)u'(t) + q(t)u(t) = g_1(t)\) and \(v''(t) + p(t)v'(t) + q(t)v(t) = g_2(t)\), then a linear combination of these functions \(y_p(t) = a_1u(t) + a_2v(t)\) is a particular solution to the third differential equation \(y''(t) + p(t)y'(t) + q(t)y(t) = a_1g_1(t) + a_2g_2(t)\).

Step-by-step solution

Given Information and Proposed Particular Solution

We are given two functions u(t) and v(t) that are solutions to the following differential equations: $$ u^{\prime \prime}+p(t) u^{\prime}+q(t) u=g_{1}(t) \quad \text { and } \quad v^{\prime \prime}+p(t) v^{\prime}+q(t) v=g_{2}(t) $$ The proposed particular solution is of the form: $$ y_{p}(t)=a_{1} u(t) + a_{2} v(t) $$

Finding the First Derivative of \(y_p(t)\)

To find the first derivative of \(y_p(t)\), we will use the linearity of differentiation and the constant multiple rule. $$ y_p'(t) = \frac{d}{dt}(a_1u(t) + a_2v(t)) = a_1\frac{d}{dt}u(t) + a_2\frac{d}{dt}v(t) = a_1u'(t) + a_2v'(t) $$

Finding the Second Derivative of \(y_p(t)\)

We will now find the second derivative of \(y_p(t)\), again using the constant multiple rule and the linearity of differentiation: $$ y_p''(t) = \frac{d^2}{dt^2}(a_1u(t) + a_2v(t)) = a_1\frac{d^2}{dt^2}u(t) + a_2\frac{d^2}{dt^2}v(t) = a_1u''(t) + a_2v''(t) $$

Plugging the Derivatives Into the Third Differential Equation

We will now check if our proposed function \(y_p(t)\) is a solution to the third differential equation, by plugging in its derivatives. $$ y^{\prime \prime}+p(t) y^{\prime}+q(t) y = (a_1u''(t) + a_2v''(t)) + p(t)(a_1u'(t) + a_2v'(t)) + q(t)(a_1u(t) + a_2v(t)) $$

Distribute the Terms and Rearrange

Now we will distribute the terms and rearrange them to check if they are equal to the right-hand side of the third differential equation. The given right-hand side is: $$ a_1g_1(t) + a_2g_2(t) $$ Distributing and rearranging the left-hand side, we get: $$ a_1(u''(t) + p(t)u'(t) + q(t)u(t)) + a_2(v''(t) + p(t)v'(t) + q(t)v(t)) $$ Using the given information that \(u(t)\) and \(v(t)\) are solutions to their respective differential equations, we can replace the expressions inside the parentheses with \(g_1(t)\) and \(g_2(t)\): $$ a_1g_1(t) + a_2g_2(t) $$ Since both sides are equal, our proposed function \(y_p(t)\) is indeed a particular solution to the third differential equation.

Key concepts

Differential Equations

Differential equations play a vital role in understanding dynamics and change, describing a quantity's rate of change in terms of another. They involve derivatives, expressions showing how one variable, typically time, affects another.

In this exercise, we're given two differential equations involving functions \( u(t) \) and \( v(t) \), which are subject to conditions such as \( u''(t) + p(t)u'(t) + q(t)u(t) = g_1(t) \) and \( v''(t) + p(t)v'(t) + q(t)v(t) = g_2(t) \). The task involves proving a specific functional form, \( y_p(t) = a_1 u(t) + a_2 v(t) \), satisfies a distinct differential equation.

Solving such equations usually involves finding a general solution or specific particular solutions by applying initial values or boundary conditions. Differential equations can represent various phenomena, making them central tools in fields such as physics, engineering, and economics.

Understanding and confirming solutions enhances problem-solving skills around dynamic systems.

Linearity of Differentiation

Linearity of differentiation is a fundamental property used when dealing with differential equations. This principle refers to the fact that the differentiation operation respects addition and scalar multiplication. Essentially, it means

• The derivative of a sum is the sum of the derivatives
• The derivative of a constant times a function is the constant times the derivative of the function

Using these properties, we easily differentiate composite functions like \( y_p(t) = a_1 u(t) + a_2 v(t) \).

For instance, in Step 2, this property allowed us to take the derivative of the sum and manage constants separately: \( y'_p(t) = a_1 u'(t) + a_2 v'(t) \). When calculating up to second derivatives, these principles remain applicable and indispensable, offering simplicity and clarifying complex expressions.

Through linearity, we can extend derivative operations over sums and differences without altering the fundamental properties of original functions.

Constant Multiple Rule

Parts of calculus, such as differentiation, involve rules like the constant multiple rule. This rule states that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.

Consider our function \(y_p(t) = a_1 u(t) + a_2 v(t)\). When computing \(y_p'(t)\), each constant \(a_1\) and \(a_2\) remains unaffected, allowing us to easily transform: \( y'_p(t) = a_1 u'(t) + a_2 v'(t) \). This simplification is vital in verifying or computing further derivatives such as \(y''_p(t)\).
The constant multiple rule is not only conceptually integral but computationally freeing. This principle is a building block in calculus, streamlining calculus operations when constants multiply functions. It ensures that constants shift with derivatives but do not complicate the operations' governing mechanisms, emphasizing the roses of differentiation.

Superposition Principle

The superposition principle is a key concept in solving linear differential equations. This principle states that if two functions are solutions to a linear differential equation, any linear combination of these functions is also a solution. This highlights the heavily interconnected nature of linear differential systems.

In this exercise, establishing that \(y_p(t) = a_1 u(t) + a_2 v(t)\) is a solution reflects the superposition principle. Given that \(u(t)\) and \(v(t)\) are solutions to their respective differential equations, any weighted sum (considering weights \(a_1\) and \(a_2\)) of these functions remains a solution to a similarly structured equation.
This principle facilitates leveraging known solutions to generate new solutions in linear contexts. Its importance in applied mathematics and physics is unmatched, enabling the construction of complex solutions from simpler, individual components. Through superposition, solutions in engineering, quantum mechanics, and signal processing become intuitive and achievable.