Solve the following differential equation: (1 + x)*(dy/dx) = y
where y is a funtion of X
solve by the following:
A)series expansion
B) elemantary method
please show working
To solve the given differential equation (1 + x)(dy/dx) = y, let's solve it using both series expansion and the elementary method.
A) Series Expansion Method:
Step 1: Let's assume a power series solution of the form y(x) = ∑(n=0 to ∞) aₙxⁿ.
Step 2: Differentiate y(x) with respect to x:
dy/dx = ∑(n=0 to ∞) aₙ * d/dx(xⁿ)
= ∑(n=0 to ∞) aₙ * nxⁿ⁻¹
Step 3: Substitute the power series solution and its derivative back into the original differential equation:
(1 + x) * (∑(n=0 to ∞) aₙ * nxⁿ⁻¹) = ∑(n=0 to ∞) aₙxⁿ
Step 4: Simplify the equation:
∑(n=0 to ∞) aₙ * nxⁿ⁺¹ + ∑(n=0 to ∞) aₙ * nxⁿ = ∑(n=0 to ∞) aₙxⁿ
Step 5: Group terms with the same power of x:
a₀ * x⁰ + ∑(n=1 to ∞) (aₙ * n * xⁿ + aₙ * xⁿ) = ∑(n=0 to ∞) aₙxⁿ
Step 6: Equate the coefficients of x for each power n:
For n = 0: a₀ = a₀
For n > 0: aₙ * n * xⁿ + aₙ * xⁿ = aₙxⁿ
Step 7: Solve for an in terms of an-1:
For n > 0: aₙ * n = aₙ
aₙ = aₙ₋₁ / n
Step 8: Write the general solution by substituting the values of an into the power series solution:
y(x) = a₀ * x⁰ + a₁ * x¹ + a₂ * x² + ...
B) Elementary Method:
To solve the differential equation using the elementary method, we can rewrite the original equation in standard form by dividing both sides by y:
(dy/dx) / y = 1 / (1 + x)
Step 1: Integrate both sides with respect to x:
∫ (dy/dx) / y dx = ∫ 1 / (1 + x) dx
Step 2: Integrate the left side using the natural logarithm property:
ln|y| = ln|1 + x| + C₁
Step 3: Simplify the equation by exponentiating both sides:
|y| = e^(ln|1 + x| + C₁)
Step 4: Remove the absolute value and express the solution in exponential form:
y = ± e^(ln|1 + x| + C₁)
Step 5: Combine the constants of integration:
y = ± e^(ln|1 + x|) * e^(C₁)
Step 6: Rewrite the constant term:
y = ± e^(ln|1 + x|) * C₂
Step 7: Combine the ± sign with the constant:
y = C₂ * (1 + x)
So, the general solution to the differential equation is y = C₂ * (1 + x), where C₂ is the arbitrary constant.