Solve the following differential equation: (1 + x)*(dy/dx) = y
where y is a funtion of X
solve by the following by:
A)series expansion
B) elemantary method
please show working
To solve the given differential equation (1 + x)(dy/dx) = y, we will use both the series expansion method and the elementary method.
A) Series Expansion Method:
Step 1: Assume a power series solution of the form y = ∑(n=0 to ∞)a_n*x^n, where a_n represents the coefficients of the power series.
Step 2: Calculate the derivatives involved in the differential equation:
- First derivative: dy/dx = ∑(n=0 to ∞)a_n(n)x^(n-1) = ∑(n=1 to ∞)a_nx^(n-1)
- Second derivative: d^2y/dx^2 = ∑(n=1 to ∞)a_n(n)(n-1)x^(n-2)
Step 3: Substitute the power series solution and its derivatives into the differential equation:
(1 + x)(∑(n=1 to ∞)a_nx^(n-1)) = ∑(n=0 to ∞)a_nx^n
Step 4: Multiply out the terms and rearrange them based on powers of x:
∑(n=1 to ∞)[(n-1)a_n + a_n]x^n + ∑(n=0 to ∞)a_nx^n = ∑(n=0 to ∞)a_nx^n
Step 5: Equate the coefficients of each term to zero:
For n = 0, we have: a_0 = 0.
For n ≥ 1, we get the recurrence relation: (n-1)a_n + a_n = a_n-1
Step 6: Solve the recurrence relation recursively to find the coefficients a_n:
For n = 1, we have: a_1 = a_0/0! = 0.
For n = 2, we have: a_2 = a_1/1! = 0.
For n = 3, we have: a_3 = a_2/2! = 0.
Continuing this pattern, we find that a_n = 0 for all n ≥ 1.
Step 7: Substitute the coefficients into the power series solution:
y = 0 + 0*x + 0*x^2 + ... = 0
Therefore, the solution to the given differential equation using the series expansion method is y = 0.
B) Elementary Method:
Step 1: Rearrange the differential equation by dividing both sides by (1 + x):
(dy/dx) = y / (1 + x)
Step 2: Separate the variables by multiplying both sides by dx and dividing by y:
dy / y = dx / (1 + x)
Step 3: Integrate both sides:
∫(dy / y) = ∫(dx / (1 + x))
Step 4: Evaluate the integrals:
ln|y| = ln|1 + x| + C, where C is the constant of integration.
Step 5: Apply the exponential function to both sides to remove the natural logarithms:
e^(ln|y|) = e^(ln|1 + x| + C)
y = e^(ln|1 + x|) * e^C
Step 6: Simplify the expression by using the properties of exponents and absorb the constant into a new constant:
y = C*(1 + x)
Therefore, the solution to the given differential equation using the elementary method is y = C*(1 + x), where C is an arbitrary constant.