What is the particular solution to the differential equation start fraction, d, y, divided by, d, x, end fraction, equals, secant, squared, left bracket, x, right bracket, left bracket, 1, minus, 3, y, right bracket dxdy =sec 2 (x)(1−3y) with the initial condition y, left bracket, start fraction, pi, divided by, 4, end fraction, right bracket, equals, 0, question marky( 4π )=0?
Question
What is the particular solution to the differential equation start fraction, d, y, divided by, d, x, end fraction, equals, secant, squared, left bracket, x, right bracket, left bracket, 1, minus, 3, y, right bracket dxdy =sec 2 (x)(1−3y) with the initial condition y, left bracket, start fraction, pi, divided by, 4, end fraction, right bracket, equals, 0, question marky( 4π )=0?
Solution
The differential equation given is dy/dx = sec^2(x)(1-3y) with the initial condition y(π/4) = 0.
To solve this, we can use the method of separation of variables.
Step 1: Separate the variables by dividing both sides by (1-3y) and multiplying both sides by dx. This gives us:
1/(1-3y) dy = sec^2(x) dx
Step 2: Integrate both sides:
∫1/(1-3y) dy = ∫sec^2(x) dx
The integral of 1/(1-3y) with respect to y is -(1/3)ln|1-3y| and the integral of sec^2(x) with respect to x is tan(x). So we have:
-(1/3)ln|1-3y| = tan(x) + C
Step 3: Solve for y:
ln|1-3y| = -3tan(x) - 3C
1-3y = e^(-3tan(x) - 3C)
y = (1 - e^(-3tan(x) - 3C))/3
Step 4: Apply the initial condition y(π/4) = 0:
0 = (1 - e^(-3tan(π/4) - 3C))/3
Solving for C gives C = ln(1/3).
So the particular solution to the differential equation is:
y = (1 - e^(-3tan(x) - ln(1/3)))/3
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