Question

To solve the differential equation ly′′+my′+fy=elx with the given initial conditions y(0)=m and y′(0)=f, you can use the method of undetermined coefficients.

Answer 1

The solution to the differential equation ly'' + my' + fy = e^lx with initial conditions y(0) = m and y'(0) = f using the method of undetermined coefficients is given by:

`\[ y(x) = e^{(-m)/(2l)x} \left( A\cos\left((√(4lf - m^2))/(2l)x\right) + B\sin\left((√(4lf - m^2))/(2l)x\right) \right) + \frac{e^{(m)/(2l)x}}{l} \]`

Step-by-step explanation:

To solve the differential equation, we first find the complementary solution by assuming
`\(y_c(x) = e^{(-m)/(2l)x}(A\cos((√(4lf - m^2))/(2l)x) + B\sin((√(4lf - m^2))/(2l)x))\)`, where A and B are constants. Then, we find the particular solution by assuming
`\(y_p(x) = \frac{e^{(m)/(2l)x}}{l}\)` due to the term
`\(e^{\frac{lx}}{l}\)` on the right-hand side. The general solution is the sum of the complementary and particular solutions, and we use the initial conditions to determine the values of A and B. Finally, substituting these values back into the general solution gives the specific solution to the given initial value problem.

The method of undetermined coefficients involves assuming a form for the particular solution based on the non-homogeneous term and solving for the undetermined coefficients. In this case, the termin the right-hand side suggests a particular solution in the form
`\(\frac{e^{(m)/(2l)x}}{l}\).`The constants A and B are determined by the initial conditions, providing a unique solution to the given differential equation and initial value problem.