Question

Suppose \(p_{1}, p_{2}, q_{1},\) and \(q_{2}\) are continuous on \((a, b)\) and the equations $$y^{\prime \prime}+p_{1}(x) y^{\prime}+q_{1}(x) y=0 \quad \text { and } \quad y^{\prime \prime}+p_{2}(x) y^{\prime}+q_{2}(x) y=0$$ have the same solutions on \((a, b) .\) Show that \(p_{1}=p_{2}\) and \(q_{1}=q_{2}\) on \((a, b) .\) Hint: Use Abel's formula.

Short answer

Question: Prove that if two second-order linear differential equations have the same solutions on the interval \((a, b)\), then their coefficients are equal on that interval. Answer: To prove that the coefficients of two second-order linear differential equations are equal on the interval \((a, b)\) if they have the same solutions on that interval, we applied Abel's formula, compared their Wronskians, and investigated the relationship between their coefficients. By showing that \(p_1(x) = p_2(x)\) and \(q_1(x) = q_2(x)\) on \((a, b)\), we demonstrated that their coefficients are indeed equal on that interval.

Step-by-step solution

Apply Abel's formula to both differential equations

Let \(y_1\) and \(y_2\) be the common solutions to both given differential equations. Then, we can use Abel's formula for each equation: For the first equation: $$W_{1}(y_1, y_2) = \text{const}_1 e^{- \int p_{1}(x) dx}$$ For the second equation: $$W_{2}(y_1, y_2) = \text{const}_2 e^{- \int p_{2}(x) dx}$$ Since both equations have the same solutions, their Wronskians will also be equal, i.e., \(W_{1}(y_1, y_2) = W_{2}(y_1, y_2)\).

Compare Wronskians and find the relationship between \(p_1\) and \(p_2\)

We can equate \(W_1\) and \(W_2\) to find: $$\text{const}_1 e^{- \int p_{1}(x) dx} = \text{const}_2 e^{- \int p_{2}(x) dx}$$ Divide both sides by \(\text{const}_1\): $$e^{- \int p_{1}(x) dx} = \frac{\text{const}_2}{\text{const}_1} e^{- \int p_{2}(x) dx}$$ Take the logarithm of both sides: $$- \int p_{1}(x) dx = \ln{\frac{\text{const}_2}{\text{const}_1}} - \int p_{2}(x) dx$$ Differentiate both sides: $$- p_{1}(x) = - p_{2}(x)$$ Hence, we have shown that \(p_{1}(x) = p_{2}(x)\), on \((a, b)\).

Investigate the relationship between \(q_1\) and \(q_2\)

Now that we have shown that \(p_{1}(x) = p_{2}(x)\), we can rewrite both differential equations using a single function \(p(x)\): $$y^{\prime \prime}+p(x)y^{\prime}+q_{1}(x)y=0 \quad \text{and} \quad y^{\prime \prime}+p(x)y^{\prime}+q_{2}(x)y=0$$ Subtract the second equation from the first equation: $$q_{1}(x) y - q_{2}(x) y = 0$$ Both \(y_1\) and \(y_2\) are solutions to both differential equations, so they are nonzero on \((a, b)\). Therefore, we can divide both sides by \(y\): $$q_{1}(x) - q_{2}(x) = 0$$ Finally, we have shown that \(q_{1}(x) = q_{2}(x)\), on \((a, b)\). In conclusion, we have demonstrated that if two differential equations have the same solutions on an interval \((a, b)\), then their coefficients \(p_1 = p_2\) and \(q_1 = q_2\) on that interval.

Key concepts

Abel's Formula

Abel's formula is an important tool used in the study of differential equations. It allows us to determine the Wronskian of solutions to a second-order linear homogeneous differential equation given in the form:

• \( y'' + p(x)y' + q(x)y = 0 \)

The formula is expressed as:

• \( W(y_1, y_2) = c \, e^{-\int p(x) \, dx} \)

where \( y_1 \) and \( y_2 \) are solutions of the differential equation, \( c \) is a constant, and \( W(y_1, y_2) \) represents the Wronskian of the solutions. This formula is powerful because it tells us that the Wronskian only depends on the coefficient \( p(x) \) and not on \( q(x) \). Therefore, if two differential equations have the same solutions, their Wronskians must be equal, helping us deduce relationships between the coefficients of the equations.
In the exercise problem, using Abel's formula resulted in showing that \( p_1(x) = p_2(x) \) when the solutions were the same for both equations on an interval \((a, b)\). This illustrates the utility of Abel's Formula in exploring the equivalence of the coefficients beyond simple similarity of solutions.

Wronskian

The Wronskian is a determinant associated with a set of functions, usually solutions to a system of differential equations. It is used to determine whether a set of functions is linearly independent. For two functions, \( y_1 \) and \( y_2 \), the Wronskian is given by:

• \( W(y_1, y_2) = y_1 y_2' - y_2 y_1' \)

If \( W(y_1, y_2) \) is non-zero on an interval, the functions \( y_1 \) and \( y_2 \) are linearly independent over that interval. In the context of the exercise, the Wronskians of two sets of solutions were equated using Abel's formula, showing that the differential equations must have identical \( p \) coefficients. The problem shows that if the Wronskians are equal or are always non-zero, the solution sets describe the same system, backing up the coefficient equality conclusion achieved by Abel's formula.
Thus, the Wronskian proved critical in proving the equivalency of the two differential equations whose solutions were otherwise indistinguishable over a given interval.

Continuous Functions

Continuous functions are the backbone of calculus and are critical when dealing with differential equations. A function \( f(x) \) is continuous on an interval \((a, b)\) if, roughly speaking, you can draw the graph of the function on that interval without lifting your pencil. This means there are no jumps, holes, or breaks. In mathematical terms, a function is continuous at a point \( c \) if:

• \( \lim_{x \to c} f(x) = f(c) \)

For differential equations described in the exercise, the coefficients \( p_1, p_2, q_1, \) and \( q_2 \) are continuous on the given interval \((a, b)\).
This is an important condition because many mathematical tools, including Abel's formula used in the solution exercise, require the functions to be continuous to ensure that integral calculations are valid. In the context of solving the given differential equations, the continuity of the coefficient functions makes both theoretical derivations and practical computations feasible without involving discontinuities issues.
Continuous functions behave predictably, meaning they are suitable for finding precise solutions to differential equations by established methods like Abel's formula.