10(t+1)dydt−9y=9t,

This is a first order linear differential equation. We can solve it using an integrating factor.

Step 1: Rewrite the equation in standard form The standard form of a first order linear differential equation is dy/dt + p(t)y = g(t). So, we rewrite the given equation as:

dy/dt - (9/10(t+1))y = 9t/10(t+1)

Step 2: Find the integrating factor The integrating factor is e^(∫p(t) dt). So, we need to compute the integral of -9/10(t+1). This is -9/10 * ∫dt/(t+1) = -9/10 * ln|t+1|. Therefore, the integrating factor is e^(-9/10 * ln|t+1|) = (t+1)^(-9/10).

Step 3: Multiply through by the integrating factor This gives us (t+1)^(-9/10) dy/dt - 9/10(t+1)^(-19/10)y = 9t/10(t+1)^(-9/10).

Step 4: The left hand side is now the derivative of (t+1)^(-9/10)y with respect to t. So, we can write the equation as d/dt[(t+1)^(-9/10)y] = 9t/10(t+1)^(-9/10).

Step 5: Integrate both sides with respect to t ∫d/dt[(t+1)^(-9/10)y] dt = ∫9t/10(t+1)^(-9/10) dt.

This gives us (t+1)^(-9/10)y = ∫9t/10(t+1)^(-9/10) dt + C, where C is the constant of integration.

Step 6: Solve for y Finally, we solve for y to get the solution of the differential equation:

y = (t+1)^(9/10) * [∫9t/10(t+1)^(-9/10) dt + C].