How to Avoid Recurrence of h(x,t) on Both Sides of Equation 7 in ODE Analysis?

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The discussion focuses on the challenge of avoiding the recurrence of h(x,t) on both sides of Equation 7 in the analysis of ordinary differential equations (ODEs). The transformation of h(x,t) using the Fourier transform is established, leading to a series of equations (Eq 1 to Eq 7) that ultimately reveal the issue of h(x,t) appearing on both sides of Eq 7. The participants seek methods to modify the solution of Eq 5 to eliminate this redundancy in the final equation.

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Let’s consider Uc to be transformed form of h(x,t) by applying Fourier transform
Then solution of Eq 1 by integrating factor is as in Eq 2
And by applying on Eq 2 inverse Fourier transform & some simplification gives us final solution as Eq 3
But what if f(t) in Eq 1 is equal to Eq 4
Putting value of f(t) from Eq 4 in Eq 1 gives us Eq 5
Similarly by integrating factor solution of Eq 5 comes out to be as in Eq 6
Applying Fourier inverse on Eq 6 gives final solution as Eq 7
The problem is in final equation (Eq 7)I have h(x,tow) on right hand side which is actually same what I want to find out and is on left side of eq 7. How can I modify solution of Eq 5 that h(x,tow) in Eq 6 does not remain on right hand side?

Equations are in following link
https://www.physicsforums.com/attachments/equations-png.95912/
 
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