(b) Find a particular solution of the differential equation
(x+1)dxdy=2e−y−1, given that y=0 when x=0.
(differential equation, particular solution, initial condition, separable differential equation)
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Step-by-Step Solution
Step 1: Separate the variables
The given differential equation is a variable separable type. To solve it, we need to rearrange the terms so that all y terms are on one side with dy, and all x terms are on the other side with dx. We start by dividing both sides by (2e−y−1) and (x+1) and multiplying both sides by dx.
Step 2: Integrate both sides
Now that the variables are separated, we integrate both sides of the equation. This step introduces an integration constant, which we will solve for later using the initial condition.
Step 3: Simplify the left integral
To integrate the left side, we multiply the numerator and denominator by ey. This transforms the denominator into a simpler form for substitution, making the integral easier to solve.
Step 4: Substitute for the left integral
We use a substitution method for the left integral. Let u=2−ey. Then, differentiate u with respect to y to find du. This substitution allows us to express the integral in terms of u.
Step 5: Evaluate the left integral
Substituting u and du into the left integral gives us ∫u−du. This is a standard integral, which evaluates to −ln∣u∣. Substituting back u=2−ey, we get −ln∣2−ey∣+C1.
Step 6: Evaluate the right integral
The right integral ∫x+11dx is also a standard integral. It evaluates to ln∣x+1∣+C2. We'll combine the constants of integration later.
Step 7: Combine the integrals
Equating the results of both integrals, we get −ln∣2−ey∣=ln∣x+1∣+C, where C=C2−C1 is the combined constant of integration. This is the general solution to the differential equation.
Step 8: Apply initial condition
We are given the initial condition y=0 when x=0. Substitute these values into the general solution to find the specific value of the constant C. Since e0=1 and ln(1)=0, we find that C=0.
Step 9: Substitute C back into the solution
Now that we have determined C=0, we substitute this value back into the general solution to obtain the particular solution. This equation implicitly defines y in terms of x.
Step 10: Simplify and solve for y (optional)
To express y explicitly, we can perform algebraic manipulations. We move the negative sign into the logarithm using the property alnb=lnba. Then, we equate the arguments of the logarithms and solve for ey. Taking the natural logarithm of both sides gives us y as a function of x.