Find the particular solution of the differential equation
y'=(5x+21y)/7x
satisfying the initial condition y(1)=5.
To find the particular solution of the differential equation, we can use the method of separation of variables. Here's how to do it step by step:
Step 1: Rewrite the differential equation in the form dy/dx = f(x)g(y), where f(x) = (5x + 21y) and g(y) = 1/(7x).
Step 2: Separate the variables by multiplying both sides of the equation by dx and dividing both sides by g(y):
(1/g(y)) dy = f(x) dx
Step 3: Integrate both sides of the equation with respect to their respective variables:
∫ (1/g(y)) dy = ∫ f(x) dx
To find the integral of (1/g(y)), we need to rearrange g(y) as follows:
g(y) = 1/(7x) = x^(-1) * (7^(-1)) = (1/7) * x^(-1)
Now, we can integrate both sides of the equation:
∫ (1/(1/7 * x^(-1))) dy = ∫ (5x + 21y) dx
Step 4: Simplify and solve the integrals:
7 ∫ x dy = ∫ (5x + 21y) dx
Step 5: Integrate both sides:
7 ∫ x dy = ∫ (5x + 21y) dx
y = (5/2)x^2 + 21yx + C
Here, C is the constant of integration.
Step 6: Apply the initial condition y(1) = 5 to find the value of the constant C:
5 = (5/2)(1^2) + 21(1)(5) + C
5 = 5/2 + 105 + C
C = -5/2 - 105 + 5
C = -97/2
Step 7: Substitute the value of C back into the solution:
y = (5/2)x^2 + 21yx - 97/2
Therefore, the particular solution of the differential equation y' = (5x + 21y)/(7x) satisfying the initial condition y(1) = 5 is:
y = (5/2)x^2 + 21yx - 97/2