Chapter 2: Problem 13

For the problem \(y^{\prime \prime}-k^{2} y=f(x), y(0)=0, y^{\prime}(0)=1\), a. Find the initial value Green's function. b. Use the Green's function to solve \(y^{\prime \prime}-y=e^{-x}\). c. Use the Green's function to solve \(y^{\prime \prime}-4 y=e^{2 x}\).

Short Answer

Expert verified

The Green's function for the given differential equation is derived first, then it is used as an integral kernel to solve the other two differential equations. The Green's function is an ingenious tool that turns a non-homogeneous differential equation into an integral equation which can be more straightforward to solve.

Step by step solution

Finding the Green's Function

The Green's function for the differential equation \(y^{\prime \prime}-k^{2} y=0\) is found by solving the homogeneous equation \(y^{\prime \prime}-k^{2} y=0\) for two cases, that is, \(y_{1}(x)\) for \(y(0)=0\) and \(y^{\prime}(0)=1\) and \(y_{2}(x)\) for \(y(0)=1\) and \(y^{\prime}(0)=0\), and then using this general form of green's function \(G(x, \xi)=\frac{y_{1}(\xi) y_{2}(x)-y_{1}(x) y_{2}(\xi)}{W[\xi]}\), where \(W[\xi]\) is the wronskian of \(y_{1}\) and \(y_{2}\).

Applying the Green's Function to Solve the second differential equation

With the Green's function at disposal, apply it directly to the second differential equation. The solution for the equation can be written as: \(y(x) =\int_{0}^{\infty} G(x, \xi) f(\xi) d\xi\), where \(f(x)=e^{-x}\) and \(G(x, \xi)\) is the Green's function found in step 1.

Applying the Green's Function to Solve the third differential equation

Similarly, proceed to apply the Green's function to solve the third differential equation by following the method of step 2. So, the solution for this equation can be written as: \(y(x) =\int_{0}^{\infty} G(x, \xi) f(\xi) d\xi\), where \(f(x)=e^{2x}\) and \(G(x, \xi)\) is the Green's function found in step 1.